Skip to main content

A new version of \(( p,q ) \)-Hermite–Hadamard’s midpoint and trapezoidal inequalities via special operators in \(( p,q ) \)-calculus

Abstract

In this paper, we conduct a research on a new version of the \(( p,q ) \)-Hermite–Hadamard inequality for convex functions in the framework of postquantum calculus. Moreover, we derive several estimates for \((p,q)\)-midpoint and \((p,q)\)-trapezoidal inequalities for special \(( p,q ) \)-differentiable functions by using the notions of left and right \((p,q ) \)-derivatives. Our newly obtained inequalities are extensions of some existing inequalities in other studies. Lastly, we consider some mathematical examples for some \((p,q)\)-functions to confirm the correctness of newly established results.

1 Introduction

The brilliant results of Charles Hermite and Jacques Hadamard’s studies, which ended in Hermite–Hadamard inequality, commonly known as Hadamard’s inequality, indicate the fact that if \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) is convex, we have the following double inequality:

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{\omega -\nu} \int _{\nu}^{\omega}\hbar ( x )\,dx \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$
(1)

When ħ is a concave mapping, the inequality holds in the opposite direction. There has been much research done in the Hermite–Hadamard direction for different kinds of convexities. For example, in [1, 2] the authors established some inequalities linked with midpoint and trapezoidal formulas of numerical integration for convex functions. For more results related to the above inequality and convex functions, the reader can consult [37]. There are many generalizations of convex functions, like h-convex functions, preinvex functions, m-convex functions, harmonically convex functions, \(( \alpha ,m ) \)-convex functions, convexity with respect to a pair of functions, etc. These kinds of convexities have a very large role in functional analysis, optimization theory, approximation theory, and fractional mathematical modeling [824].

Due to its immediate applications in numerical integration, probability theory, information theory, and integral operator theory, the Hermite–Hadamard inequality is of utmost significance. The field of inequalities has seen a tremendous flow of this remarkable inequality and related outstanding (Hadamard-type) inequalities over the past millennium. These inequalities are partially inspired by the results mentioned above but perhaps even more, so by the difficulty of conducting research in a variety of mathematical subdisciplines like mathematical programming, control theory, variational methods, operation research, probability, and statistics.

Besides this, quantum and postquantum calculus are very important branches of calculus having a vast range of applications in the fields of mathematics and physics. Because of numerous applications of quantum calculus (shortly, q-calculus) and postquantum calculus (shortly, \(( p,q ) \)-calculus) without limit calculus, many researchers began working on them and applying their concepts in differential equations, integral equalities, mathematical modeling, and integral inequalities [2532].

Alp et al. [33] and Bermudo et al. [34] used q-integrals to prove two different versions of q-Hermite–Hadamard inequalities along with some relevant estimates. The q-Hermite–Hadamard inequalities are described as follows.

Theorem 1.1

([33, 34])

For a convex map \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \), we have the inequalities

$$\begin{aligned}& \hbar \biggl( \frac{q\nu +\omega}{ [ 2 ] _{q}} \biggr) \leq \frac{1}{\omega -\nu}\int _{\nu}^{\omega}\hbar ( x ) \,{}_{\nu}d_{q}x \leq \frac{q\hbar ( \nu ) +\hbar ( \omega ) }{ [ 2 ] _{q}}, \end{aligned}$$
(2)
$$\begin{aligned}& \hbar \biggl( \frac{\nu +q\omega}{ [ 2 ] _{q}} \biggr) \leq \frac{1}{\omega -\nu}\int _{\nu}^{\omega}\hbar ( x ) \,{}^{\omega}d_{q}x \leq \frac{\hbar ( \nu ) +q\hbar ( \omega ) }{ [ 2 ] _{q}}. \end{aligned}$$
(3)

Remark 1.2

It is very easy to observe that by adding (3) and (4) we derive the q-Hermite–Hadamard inequality (see [34])

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{2 ( \omega -\nu ) } \biggl[ \int _{\nu}^{\omega} \hbar ( x ) \,{}_{\nu}d_{q}x+ \int _{\nu}^{\omega}\hbar ( x ) \,{}^{\omega}d_{q}x \biggr] \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$
(4)

Recently, Ali et al. [35] and Sitthiwirattham et al. [36] used new techniques to prove the following two different and new versions of Hermite–Hadamard-type inequalities in the context of q-operators.

Theorem 1.3

([35, 36])

For a convex map \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \), we have the inequalities

$$\begin{aligned} &\hbar \biggl( \frac{\nu +\omega}{2} \biggr)\leq \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x ) {}^{\frac{\nu +\omega}{2}}d_{q}x+ \int _{ \frac{\nu +\omega}{2}}^{\omega}\hbar ( x ) {}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}, \end{aligned}$$
(5)
$$\begin{aligned} &\hbar \biggl( \frac{\nu +\omega}{2} \biggr)\leq \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\nu}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{ \omega}\hbar ( x ) \,{}^{\omega}d_{q}x \biggr] \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. \end{aligned}$$
(6)

Remark 1.4

When \(q\rightarrow 1^{-}\) in (3)–(7), we recapture the traditional Hermite–Hadamard inequality (1).

Kunt et al. [37] and Vivas-Cortez et al. [38] extended the previous studies and derived several Hermite–Hadamard-type inequalities with new structures for convex functions using the \(( p,q ) \)-integrals.

Theorem 1.5

([37, 38])

For a convex mapping \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \), we have the inequalities

$$ \hbar \biggl( \frac{q\nu +p\omega}{ [ 2 ] _{p,q}} \biggr) \leq \frac{1}{p ( \omega -\nu ) } \int _{\nu}^{\omega}\hbar ( x ) \,{}_{ \nu}d_{p,q}x \leq \frac{q\hbar ( \nu ) +p\hbar ( \omega ) }{ [ 2 ] _{p,q}} $$
(7)

and

$$ \hbar \biggl( \frac{p\nu +q\omega}{ [ 2 ] _{p,q}} \biggr) \leq \frac{1}{p ( \omega -\nu ) } \int _{\nu}^{\omega}\hbar ( x ) \,{}^{ \omega}d_{p,q}x \leq \frac{p\hbar ( \nu ) +q\hbar ( \omega ) }{ [ 2 ] _{p,q}}. $$
(8)

Remark 1.6

It is also very easy to observe that by adding (7) and (8) we obtain the \(( p,q ) \)-Hermite–Hadamard inequality (see [38])

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{2p ( \omega -\nu ) } \biggl[ \int _{\nu}^{\omega} \hbar ( x ) \,{}_{\nu}d_{p,q}x+ \int _{\nu}^{\omega}\hbar ( x ) \,{}^{\omega}d_{p,q}x \biggr] \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$
(9)

Remark 1.7

It is worth mentioning that inequalities (7)–(9) are generalizations of inequalities (2)–(4), respectively, and for \(p=1\), we can obtain the q-Hermite–Hadamard inequalities.

There has been much research done in the direction of q and \(( p,q ) \)-integral inequalities for different kinds of convexity. For instance, in [3740], some new midpoint and trapezoidal inequalities via q and \(( p,q ) \)-integrals were established. The authors of [4148] used q and \(( p,q ) \)-integrals and established Simpson-type inequalities for functions with different forms of convexity. For more recent inequalities in q-calculus, see [4954].

By considering such advanced level studies we consider the convexity of functions and derive a new variant of Hermite–Hadamard inequality in the setting of \(( p,q ) \)-calculus. Furthermore, we derive some new midpoint and trapezoidal type inequalities for the special class of functions called \(( p,q ) \)-differentiable convex functions in the framework of \(( p,q ) \)-calculus. We also show that our newly established results are an extension of [36], which states the novelty of our research. The results presented here can be helpful in finding the error bounds of numerical integration formulas and variety of mathematical subdisciplines like mathematical programming, control theory, variational methods, operation research, probability, and statistics.

The structure of the paper is as follows. In Sect. 2, we recall some basics of q- and \((p,q)\)-calculus. In Sect. 3, we establish a new variant of q-Hermite–Hadamard-type inequality for some special convex \((p,q)\)-functions. In Sects. 4 and 5, we derive some new midpoint and trapezoidal inequalities for q-differentiable convexity, respectively. Section 6 briefly concludes our work.

2 q- and \(( p,q ) \)-calculus

We recall some basics of quantum calculus in this section. For \(0< q< p\leq 1\), we denote [38, 55]

$$ [ n ] _{q}=\frac{1-q^{n}}{1-q} $$

and

$$ [ n ] _{p,q}=\frac{p^{n}-q^{n}}{p-q}. $$

Definition 2.1

([56])

The left or \(q_{\nu}\)-derivative of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ {}_{\nu}\mathfrak{D}_{q}\hbar ( x )= \frac{\hbar ( x ) -\hbar ( qx+ ( 1-q ) \nu ) }{ ( 1-q ) ( x-\nu ) },\quad x\neq \nu . $$
(10)

Definition 2.2

([34])

The right or \(q^{\omega}\)-derivative of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ {}^{\omega}\mathfrak{D}_{q}\hbar ( x )= \frac{\hbar ( qx+ ( 1-q ) \omega ) -\hbar ( x ) }{ ( 1-q ) ( \omega -x ) },\quad x\neq \omega . $$

Definition 2.3

([57])

The left or \(( p,q ) _{\nu}\)-derivative of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ {}_{\nu}\mathfrak{D}_{p,q}\hbar ( x )= \frac{\hbar ( px+ ( 1-p ) \nu ) -\hbar ( qx+ ( 1-q ) \nu ) }{ ( p-q ) ( x-\nu ) },\quad x \neq \nu . $$

Definition 2.4

([38])

The right or \(( p,q ) ^{\omega}\)-derivative of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ {}^{\omega}\mathfrak{D}_{p,q}\hbar ( x )= \frac{\hbar ( qx+ ( 1-q ) \omega ) -\hbar ( px+ ( 1-p ) \omega ) }{ ( p-q ) ( \omega -x ) },\quad x\neq \omega . $$

Definition 2.5

([56])

The left or \(q_{\nu}\)-integral of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ \int _{\nu}^{x}\hbar ( t ) \,{}_{\nu}d_{q}t= ( 1-q ) ( x-\nu ) \sum_{n=0}^{ \infty }q^{n} \hbar \bigl( q^{n}x+ \bigl( 1-q^{n} \bigr) \nu \bigr) . $$

Definition 2.6

([34])

The right or \(q^{\omega}\)-integral of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [\nu ,\omega ] \) is defined as

$$ \int _{x}^{\omega}\hbar ( t ) \,{}^{\omega}d_{q}t= ( 1-q ) ( \omega -x ) \sum_{n=0}^{ \infty }q^{n} \hbar \bigl( q^{n}x+ \bigl( 1-q^{n} \bigr) \omega \bigr) . $$

Definition 2.7

([57])

The left or \(( p,q ) _{\nu}\)-integral of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) is defined as

$$ \int _{\nu}^{x}\hbar ( t ) \,{}_{\nu}d_{p,q}t= ( p-q ) ( x-\nu ) \sum_{n=0}^{ \infty } \frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n}}{p^{n+1}}x+ \biggl( 1- \frac{q^{n}}{q^{n+1}} \biggr) \nu \biggr) $$

for \(x\in [ \nu ,p\omega + ( 1-p ) \nu ] \)

Definition 2.8

([38])

The right or \(q^{\omega}\)-integral of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) is defined as

$$ \int _{x}^{\omega}\hbar ( t ) \,{}^{\omega}d_{p,q}t= ( p-q ) ( \omega -x ) \sum_{n=0}^{ \infty } \frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n}}{p^{n+1}}x+ \biggl( 1- \frac{q^{n}}{p^{n+1}} \biggr) \omega \biggr) $$

for \(x\in [ p\nu + ( 1-p ) \omega ,\omega ] \)

For more properties and details about q- and \(( p,q ) \)-calculus, the reader can consult [34, 38, 5658].

3 \((p,q)\)-Hermite–Hadamard inequality

In this section, we establish a new version of Hermite–Hadamard inequality for convex functions and the special \((p,q)\)-operators defined in \(( p,q ) \)-calculus.

Theorem 3.1

Let \(\hbar : [ \nu ,\omega ] \subset \mathbb{R} \rightarrow \mathbb{R} \) be convex. Then

$$\begin{aligned} \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq & \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p \omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] \\ \leq & \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. \end{aligned}$$
(11)

Proof

The convexity of ħ implies that

$$ \hbar \biggl( \frac{x+y}{2} \biggr) \leq \frac{1}{2} \bigl[ \hbar ( x ) +\hbar ( y ) \bigr] . $$

Setting \(x=\frac{1-t}{2}\nu +\frac{1+t}{2}\omega \) and \(y=\frac{1+t}{2}\nu +\frac{1-t}{2}\omega \), we get

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{2} \biggl[ \hbar \biggl( \frac{1-t}{2}\nu +\frac{1+t}{2}\omega \biggr) +\hbar \biggl( \frac{1+t}{2}\nu + \frac{1-t}{2}\omega \biggr) \biggr] . $$
(12)

By \(( p,q ) \)-integrating (12) with respect to t on \([ 0,p ] \) we get

$$ p\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{2} \biggl[ \int _{0}^{p}\hbar \biggl( \frac{1-t}{2}\nu + \frac{1+t}{2}\omega \biggr) \,d_{p,q}t+ \int _{0}^{p}\hbar \biggl( \frac{1+t}{2}\nu + \frac{1-t}{2}\omega \biggr) \,d_{p,q}t \biggr] . $$

From Definitions 2.7 and 2.8 we have

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p \omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] . $$

Thus the first inequality in (11) is proved. We again use the convexity to prove the second inequality in (11):

$$ \hbar \biggl( \frac{1-t}{2}\nu +\frac{1+t}{2}\omega \biggr) +\hbar \biggl( \frac{1+t}{2}\nu +\frac{1-t}{2}\omega \biggr) \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$
(13)

By \(( p,q ) \)-integrating (13) with respect to t on \([ 0,p ] \) we get

$$ \int _{0}^{p}\hbar \biggl( \frac{1-t}{2}\nu + \frac{1+t}{2}\omega \biggr) \,d_{p,q}t+ \int _{0}^{p}\hbar \biggl( \frac{1+t}{2}\nu + \frac{1-t}{2}\omega \biggr) \,d_{p,q}t\leq p \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$

By applying Definitions 2.7 and 2.8 we obtain the required inequality. □

Remark 3.2

By assuming \(p=1\) in Theorem 3.1, we regain inequality (5).

Remark 3.3

By setting \(p=1\) and taking the limit \(q\rightarrow 1^{-}\) we regain the traditional Hermite–Hadamard inequality (1) for the classical convex functions.

Example 3.4

Consider the convex function \(\hbar : [ 0,1 ] \rightarrow \mathbb{R} \) defined as \(\hbar (x)=x^{2}\) with \(q=\frac{1}{3}\) and \(p=\frac{2}{3}\). Then

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) =\frac{1}{4} $$

and

$$ \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}= \frac{1}{2}. $$

On the other hand, by Definitions 2.7 and 2.8 we have

$$\begin{aligned} \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x =& \int _{\frac{1}{6}}^{\frac{1}{2}}x^{2}\,{}^{ \frac{1}{2}}d_{\frac{2}{3},\frac{1}{3}}x \\ =&\frac{1}{3}.\frac{1}{3}\sum_{n=0}^{\infty } \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \frac{1}{6}+ \biggl( 1-\frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggr) \frac{1}{2} \biggr) ^{2} \\ =&\frac{1}{24}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( 1- \biggl( \frac{1}{2} \biggr) ^{n} \biggr) ^{2} \\ =&\frac{1}{24}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( 1-2 \biggl( \frac{1}{2} \biggr) ^{n}+ \biggl( \frac{1}{2} \biggr) ^{2n} \biggr) \\ =&\frac{1}{24} \biggl[ 2-\frac{8}{3}+\frac{8}{7} \biggr] \\ =&\frac{5}{252} \end{aligned}$$

and

$$\begin{aligned} \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x =& \int _{\frac{1}{2}}^{\frac{5}{6}}x^{2}\,{}_{ \frac{1}{2}}d_{\frac{2}{3},\frac{1}{3}}x \\ =&\frac{1}{3}.\frac{1}{3}\sum _{n=0}^{\infty }\frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n}\frac{5}{6}+ \biggl( 1- \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggr) \frac{1}{2} \biggr) ^{2} \\ =&\frac{1}{6}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{1}{2}+ \frac{1}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggr) ^{2} \\ =&\frac{1}{24} \biggl[ 2+\frac{8}{3}+\frac{8}{7} \biggr] \\ =&\frac{61}{252}. \end{aligned}$$

Thus

$$\begin{aligned}& \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega }{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] \\& \quad = \frac{3}{2} \biggl[ \frac{5}{252}+\frac{61}{252} \biggr] \\& \quad = \frac{99}{252}. \end{aligned}$$

It is clear that

$$ \frac{1}{4}< \frac{99}{252}< \frac{1}{2}. $$

4 \((p,q)\)-Midpoint inequalities

In this section, we establish some new inequalities of midpoint type for \(( p,q ) \)-differentiable functions in the setting of \(( p,q ) \)-calculus. We begin with a lemma, which has a great role in establishing the inequalities of this section.

Lemma 4.1

For \(\hbar : [ \nu ,\omega ] \subset \mathbb{R} \rightarrow \mathbb{R} \), if \({}_{\nu}\mathfrak{D}_{p,q}\hbar \) and \({}^{\omega}\mathfrak{D}_{p,q}\hbar \) are continuous and integrable mappings over \([ \nu ,\omega ] \), then we have the following identity:

$$\begin{aligned}& \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega }{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \\& \quad = \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \\& \qquad {}-{}^{\omega}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr) \,d_{p,q}t \biggr] . \end{aligned}$$
(14)

Proof

Definitions 2.3 and 2.4 give

$$ {}^{\omega}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) =2p \biggl[ \frac{\hbar ( \frac{q}{p}t\nu + ( 1-\frac{q}{p}t ) \frac{\nu +\omega}{2} ) -\hbar ( t\nu + ( 1-t ) \frac{\nu +\omega}{2} ) }{ ( p-q ) ( \omega -\nu ) t} \biggr] $$
(15)

and

$$ {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) =2p \biggl[ \frac{\hbar ( t\omega + ( 1-t ) \frac{\nu +\omega}{2} ) -\hbar ( \frac{q}{p}t\omega + ( 1-\frac{q}{p}t ) \frac{\nu +\omega}{2} ) }{ ( p-q ) ( \omega -\nu ) t} \biggr] . $$
(16)

By Definition 2.8 from (15) we have

$$\begin{aligned}& \int _{0}^{p} ( 1-qt ) {}^{\omega} \mathfrak{D}_{p,q} \hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \,d_{p,q}t \\& \quad = \int _{0}^{p} ( 1-qt )p \frac{\hbar ( \frac{q}{p}t\nu + ( 1-\frac{q}{p}t ) \frac{\nu +\omega}{2} ) -\hbar ( t\nu + ( 1-t ) \frac{\nu +\omega}{2} ) }{ ( p-q ) ( \frac{\omega -\nu}{2} ) t}\,d_{p,q}t \\& \quad = \frac{2p^{2}}{\omega -\nu} \Biggl[ \sum_{n=0}^{\infty } \hbar \biggl( \frac{q^{n+1}}{p^{n+1}}\nu + \biggl( 1-\frac{q^{n+1}}{p^{n+1}} \biggr) \frac{\nu +\omega}{2} \biggr) \\& \qquad {} -\sum_{n=0}^{ \infty } \hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1-\frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \qquad {} -\frac{2p^{2}q}{\omega -\nu} \Biggl[ \sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n+1}}{p^{n+1}}\nu + \biggl( 1- \frac{q^{n+1}}{p^{n+1}} \biggr) \frac{\nu +\omega}{2} \biggr) \\& \qquad {} -\sum _{n=0}^{\infty }\frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = \frac{2p^{2}}{\omega -\nu} \biggl[ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) -\hbar ( \nu ) \biggr] \\& \qquad {}-\frac{2p^{2}q}{\omega -\nu} \Biggl[ \frac{1}{q}\sum _{n=0}^{ \infty }\frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) -\frac{1}{q} \hbar ( \nu ) \\& \qquad {}-\frac{1}{p}\sum_{n=0}^{\infty } \frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& = \frac{2p^{2}}{\omega -\nu}\hbar \biggl( \frac{\nu +\omega}{2} \biggr) -\frac{2p^{2}q}{\omega -\nu} \Biggl[ \frac{p-q}{pq}\sum_{n=0}^{\infty } \frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = \frac{2p^{2}}{\omega -\nu}\hbar \biggl( \frac{\nu +\omega}{2} \biggr) -\frac{4p}{ ( \omega -\nu ) ^{2}} \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x, \end{aligned}$$

which implies that

$$\begin{aligned}& \frac{\omega -\nu}{4p^{2}} \int _{0}^{p} ( 1-qt ) {}^{ \omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p} \omega \biggr) \,d_{p,q}t \\& \quad = \frac{1}{2}\hbar \biggl( \frac{\nu +\omega}{2} \biggr) - \frac{1}{p ( \omega -\nu ) }\int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x. \end{aligned}$$
(17)

Similarly, from Definition 2.7 and relation (16) we have

$$\begin{aligned}& \frac{\omega -\nu}{4p^{2}} \int _{0}^{p} ( 1-qt ) {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu +\frac{p+t}{2p} \omega \biggr) \,d_{p,q}t \\& \quad = \frac{1}{p ( \omega -\nu ) } \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x- \frac{1}{2}\hbar \biggl( \frac{\nu +\omega}{2} \biggr) . \end{aligned}$$
(18)

Thus we derive the required identity (14) by subtracting (17) from (18). □

Remark 4.2

In Lemma 4.1, for \(p=1\), we obtain the following identity:

$$\begin{aligned}& \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}} \hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \\& \quad = \frac{\omega -\nu}{4} \biggl[ \int _{0}^{1} ( 1-qt ) \biggl( {}_{\nu} \mathfrak{D}_{q}\hbar \biggl( \frac{1-t}{2}\nu + \frac{1+t}{2}\omega \biggr) \\& \qquad {}-{}^{\omega}\mathfrak{D}_{q}\hbar \biggl( \frac{1+t}{2}\nu +\frac{1-t}{2}\omega \biggr) \biggr) \,d_{q}t \biggr] , \end{aligned}$$

which was obtained by Sitthiwirattham et al. [36].

Theorem 4.3

If Lemma 4.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert \) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert \) are convex, then

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x )\, {}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{8p^{3}} \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q\frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr] . \end{aligned}$$
(19)

Proof

Taking the modulus in (14), by the convexity properties of \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert \) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert \), we estimate

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( \biggl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert \\& \qquad {}+ \biggl\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p} \nu + \frac{p-t}{2p}\omega \biggr) \biggr\vert \biggr) \,d_{p,q}t \biggr] \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( \frac{p-t}{2p} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert +\frac{p+t}{2p} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr) \,d_{p,q}t \\& \qquad {} + \int _{0}^{p} ( 1-qt ) \biggl( \frac{p+t}{2p} \big\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert + \frac{p-t}{2p} \big\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert \biggr) \,d_{p,q}t \biggr] \\& \quad = \frac{\omega -\nu}{8p^{3}} \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \biggl\vert {}_{\nu} \mathfrak{D}_{p,q} \hbar ( \nu ) \biggl\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert { }_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q\frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr] , \end{aligned}$$

and our proof is completed. □

Remark 4.4

If we set \(p=1\) in the previous theorem, then

$$\begin{aligned}& \biggl\vert \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega} \hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{8} \biggl[ \biggl( \frac{q}{ [ 2 ] _{q}}- \frac{q^{3}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigl\vert {}_{ \nu}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert + \biggl( \frac{2+q}{ [ 2 ] _{q}}-q \frac{ [ 3 ] _{q}+ [ 2 ] _{q}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{q} \hbar ( \omega ) \bigl\vert \\& \qquad {}+ \biggl( \frac{2+q}{ [ 2 ] _{q}}-q \frac{ [ 3 ] _{q}+ [ 2 ] _{q}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert + \biggl( \frac{q}{ [ 2 ] _{q}}- \frac{q^{3}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \big\vert {} ^{\omega} \mathfrak{D}_{q}\hbar ( \omega ) \big\vert \biggr] , \end{aligned}$$

which was obtained by Sitthiwirattham et al. [36].

Example 4.5

Consider \(\hbar : [ 0,1 ] \rightarrow \mathbb{R} \) defined by \(\hbar (x)=x^{3}\). Let also \(q=\frac{1}{3}\) and \(p=\frac{2}{3}\). Then we have the convex functions \({}_{\nu}\mathfrak{D}_{p,q}\hbar ( x ) =\frac{7x^{2}}{27}\) and \({}^{\omega}\mathfrak{D}_{p,q}\hbar ( x ) =\frac{1}{3} ( 7x^{2}+13x+7 ) \), which gives

$$\begin{aligned} \int _{pa+ ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x =& \int _{\frac{1}{6}}^{\frac{1}{2}}x^{3}\,{}^{ \frac{1}{2}}d_{\frac{2}{3},\frac{1}{3}}x \\ =&\frac{1}{3}\frac{1}{3}\sum_{n=0}^{\infty } \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \frac{1}{6}+ \biggl( 1-\frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggr) \frac{1}{2} \biggr) ^{3} \\ =&\frac{1}{48}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( 1- \biggl( \frac{1}{2} \biggr) ^{n} \biggr) ^{3} \\ =&\frac{1}{48} \biggl[ 2-4+\frac{24}{7}-\frac{16}{15} \biggr] \\ =&\frac{19}{2520} \end{aligned}$$

and

$$\begin{aligned} \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x =& \int _{\frac{1}{2}}^{\frac{5}{6}}x^{3}\,{}_{ \frac{1}{2}}d_{\frac{2}{3},\frac{1}{3}}x \\ =&\frac{1}{3}\frac{1}{3}\sum_{n=0}^{\infty } \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \frac{5}{6}+ \biggl( 1- \biggl( \frac{1}{2} \biggr) ^{n} \biggr) \frac{1}{2} \biggr) ^{3} \\ =&\frac{1}{48}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( 1+ \biggl( \frac{1}{2} \biggr) ^{n} \biggr) ^{3} \\ =&\frac{1}{48} \biggl[ 2+4+\frac{24}{7}+\frac{16}{15} \biggr] \\ =&\frac{551}{2520}. \end{aligned}$$

Thus the left-hand side of inequality (19) reduces to

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad = \biggl\vert \frac{3}{2} \biggl[ \frac{18}{2520}+ \frac{551}{2520} \biggr] -\frac{1}{8} \biggr\vert \\& \quad = \frac{9}{42}. \end{aligned}$$

On the other hand, since \({}_{\nu}\mathfrak{D}_{p,q}\hbar ( a ) =0\), \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \vert =\frac{7}{27}\), \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \vert =\frac{7}{3}\), and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \omega ) \vert =9\), The righ-hand side of inequality (19) becomes

$$\begin{aligned}& \frac{\omega -\nu}{8p^{3}} \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q\frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr] \\& \quad = \frac{27}{64} \biggl[ \biggl( \frac{8}{9}-\frac{128}{567} \biggr) \frac{7}{27}+ \biggl( \frac{8}{9}- \frac{128}{567} \biggr) \frac{7}{3}+ \biggl( 0+ \frac{16}{567} \biggr) 9 \biggr] \\& \quad = \frac{472}{567}. \end{aligned}$$

It is clear that

$$ \frac{9}{42}< \frac{472}{567}. $$

Theorem 4.6

If Lemma 4.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex for \(s\geq 1\), then

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{p^{2}+pq-qp^{2}}{ [ 2 ] _{p,q}} \biggr) ^{1-\frac{1}{s}} \\& \qquad {} \times \biggl[ \biggl( \frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert ^{s} \\& \qquad {}+\frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {} + \biggl( \frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \nu ) \big\vert ^{s} \\& \qquad {} +\frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

Proof

Taking the modulus in (14), by the power mean inequality we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( \biggl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert \\& \qquad {} + \biggl\vert {} ^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p} \nu + \frac{p-t}{2p}\omega \biggr) \biggr\vert \biggr) \,d_{p,q}t \biggr] \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \int _{0}^{p} ( 1-qt ) \,d_{p,q}t \biggr) ^{1-\frac{1}{s}} \biggl[ \biggl( \int _{0}^{p} ( 1-qt ) \biggl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \int _{0}^{p} ( 1-qt ) \bigg\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \bigg\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \biggr] . \end{aligned}$$

By the convexity of \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{p^{2}+pq-qp^{2}}{ [ 2 ] _{p,q}} \biggr) ^{1-\frac{1}{s}} \\& \qquad {} \times \biggl[ \biggl( \frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert ^{s} \\& \qquad {}+\frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {} _{\nu} \mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {} + \biggl( \frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \nu ) \big\vert ^{s} \\& \qquad {} +\frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

The proof is completed. □

Remark 4.7

In Theorem 4.6, for \(p=1\), we have the inequality

$$\begin{aligned}& \biggl\vert \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega} \hbar ( x ) \,{}_{\frac{\nu +\omega }{2}}d_{q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4} \biggl( \frac{1}{ [ 2 ] _{q}} \biggr) ^{1-\frac{1}{s}} \biggl[ \biggl( \frac{1}{2} \biggl( \frac{q}{ [ 2 ] _{q}}- \frac{q^{3}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert ^{s} \\& \qquad {}+ \frac{1}{2} \biggl( \frac{2+q}{ [ 2 ] _{q}}-q \frac{ [ 3 ] _{q}+ [ 2 ] _{q}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigl\vert {}_{ \nu}\mathfrak{D}_{q} \hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{1}{2} \biggl( \frac{2+q}{ [ 2 ] _{q}}-q \frac{ [ 3 ] _{q}+ [ 2 ] _{q}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \big\vert {} ^{\omega}\mathfrak{D}_{q}\hbar ( \nu ) \big\vert ^{s}+\frac{1}{2} \biggl( \frac{q}{ [ 2 ] _{q}}- \frac{q^{3}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr], \end{aligned}$$

which was obtained by Sitthiwirattham et al. [36].

Theorem 4.8

If Lemma 4.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex for \(s>1\), then

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{1- ( 1-pq ) ^{r+1}}{q [ r+1 ] _{p,q}} \biggr) ^{\frac{1}{r}} \\& \qquad {}\times \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert ^{s}+ \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr], \end{aligned}$$

where \(s^{-1}+r^{-1}=1\).

Proof

Taking the modulus in (14), by the Hölder inequality we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x )\, {}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( \biggl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert \\& \qquad {}+ \biggl\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p} \nu + \frac{p-t}{2p}\omega \biggr) \biggr\vert \biggr) \,d_{p,q}t \biggr] \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \int _{0}^{p} ( 1-qt )^{r}\,d_{p,q}t \biggr) ^{\frac{1}{r}} \biggl[ \biggl( \int _{0}^{p} \biggl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \\& \qquad {}+ \biggl( \int _{0}^{p} \biggl\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \biggr] . \end{aligned}$$

By the convexity of \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert ^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{1- ( 1-pq ) ^{r+1}}{q [ r+1 ] _{p,q}} \biggr) ^{\frac{1}{r}} \\& \qquad {}\times \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {} _{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert ^{s}+ \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \big\vert {} ^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

 □

Remark 4.9

In Theorem 4.8, for \(p=1\), we have the inequality

$$\begin{aligned}& \biggl\vert \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega} \hbar ( x )\, {}_{\frac{\nu +\omega }{2}}d_{q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4} \biggl( \frac{1- ( 1-q ) ^{r+1}}{q [ r+1 ] _{q}} \biggr) ^{\frac{1}{r}} \biggl[ \biggl( \frac{q}{2 ( 1+q ) } \bigl\vert {}_{\nu} \mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{2+q}{2 ( 1+q ) } \bigl\vert {}_{\nu}\mathfrak{D}_{q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{2+q}{2 ( 1+q ) } \big\vert {}^{ \omega} \mathfrak{D}_{q}\hbar ( \nu ) \big\vert ^{s}+ \frac{q}{2 ( 1+q ) } \big\vert {}^{\omega}\mathfrak{D}_{q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] , \end{aligned}$$

which was obtained by Sitthiwirattham et al. [36].

5 \((p,q)\)-Trapezoid inequalities

Now we obtain some \((p,q)\)-trapezoidal inequalities. Let us begin by the following important equality.

Lemma 5.1

For \(\hbar : [ \nu ,\omega ] \subset \mathbb{R} \rightarrow \mathbb{R} \), if \({}_{\nu}\mathfrak{D}_{p,q}\hbar \) and \({}^{\omega}\mathfrak{D}_{p,q}\hbar \) are continuous and integrable mappings over \([ \nu ,\omega ] \), then we have the identity

$$\begin{aligned}& \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x )\, {}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p \omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] \\& \quad = \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( qt ) \biggl( {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \\& \qquad {} -{}^{\omega}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr) \,d_{p,q}t \biggr]. \end{aligned}$$
(20)

Proof

By using (15) and Definition 2.8 we get

$$\begin{aligned}& \int _{0}^{p} ( qt ) {}^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \,d_{p,q}t \\& \quad = \int _{0}^{p} ( qt ) {p} \frac{\hbar ( \frac{q}{p}t\nu + ( 1-\frac{q}{p}t ) \frac{\nu +\omega}{2} ) -\hbar ( t\nu + ( 1-t ) \frac{\nu +\omega}{2} ) }{ ( p-q ) ( \frac{\omega -\nu}{2} ) t}\,d_{p,q}t \\& \quad = \frac{2p^{2}}{\omega -\nu} \Biggl[ \sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n+1}}{p^{n+1}}\nu + \biggl( 1- \frac{q^{n+1}}{p^{n+1}} \biggr) \frac{\nu +\omega}{2} \biggr) \\& \qquad {}-\sum _{n=0}^{\infty }\frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1-\frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = \frac{2p^{2}q}{\omega -\nu} \Biggl[ \frac{1}{q}\sum _{n=0}^{ \infty }\frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) -\frac{1}{q}\hbar ( \nu ) \\& \qquad {} -\frac{1}{p}\sum_{n=0}^{\infty }\frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1-\frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = \frac{2p^{2}}{\omega -\nu}\hbar ( \nu ) + \frac{2p^{2}q}{\omega -\nu} \Biggl[ \frac{p-q}{pq}\sum_{n=0}^{\infty } \frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = -\frac{2p^{2}}{\omega -\nu}\hbar ( \nu ) + \frac{4p}{ ( \omega -\nu ) ^{2}}\int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x. \end{aligned}$$

This gives

$$\begin{aligned}& \frac{\omega -\nu}{4p^{2}} \int _{0}^{p} ( qt ) {}^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \,d_{p,q}t \\& \quad = \frac{1}{p ( \omega -\nu ) } \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x- \frac{1}{2}\hbar ( \nu ) . \end{aligned}$$
(21)

Similarly, by (16) and Definition 2.7 it becomes

$$\begin{aligned}& \frac{\omega -\nu}{4p^{2}} \int _{0}^{p} ( qt ) {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \,d_{p,q}t \\& \quad = \frac{1}{2}\hbar ( \omega ) - \frac{1}{p ( \omega -\nu ) } \int _{ \frac{\nu +\omega }{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x. \end{aligned}$$
(22)

Therefore we establish the required identity (20) by equalities (21) and (22). □

Corollary 5.2

In Lemma 5.1, for \(p=1\), we obtain the new identity

$$\begin{aligned}& \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}} \hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{q}x+ \int _{ \frac{\nu +\omega}{2}}^{\omega}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \\& \quad = \frac{\omega -\nu}{4} \biggl[ \int _{0}^{1} ( qt ) \biggl( {}_{\nu} \mathfrak{D}_{q}\hbar \biggl( \frac{1-t}{2}\nu + \frac{1+t}{2}\omega \biggr) -{}^{\omega}\mathfrak{D}_{q}\hbar \biggl( \frac{1+t}{2}\nu +\frac{1-t}{2}\omega \biggr) \biggr)\, d_{q}t \biggr] . \end{aligned}$$

This identity helps us to find some estimates of q-trapezoidal inequalities.

Theorem 5.3

If Lemma 5.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert \) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert \) are convex, then

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{q ( \omega -\nu ) }{8 [ 2 ] _{p,q} [ 3 ] _{p,q}} \bigl[ \bigl( [ 3 ] _{p,q}- [ 2 ] _{p,q} \bigr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{p,q}+ [ 2 ] _{p,q} \bigr) \bigl\vert {} _{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigl\vert \\& \qquad {}+ \bigl( [ 3 ] _{p,q}+ [ 2 ] _{p,q} \bigr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{p,q}- [ 2 ] _{p,q} \bigr) \big\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar ( \omega ) \big\vert \bigr] . \end{aligned}$$
(23)

Proof

By taking the modulus in equality (20) we may write

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( qt ) \biggl( \biggl\vert {} _{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert \\& \qquad {} + \biggl\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr\vert \biggr) \,d_{p,q}t \biggr] . \end{aligned}$$
(24)

Since the functions \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert \) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert \) are convex, we have

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( qt ) \biggl( \frac{p-t}{2p} \bigl\vert {} _{\nu}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert +\frac{p+t}{2p} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr) \,d_{p,q}t \\& \qquad {} + \int _{0}^{p} ( qt ) \biggl( \frac{p+t}{2p} \big\vert {} ^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert + \frac{p-t}{2p} \big\vert {} ^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert \biggr) \,d_{p,q}t \biggr] \\& \quad = \frac{\omega -\nu}{8p^{3}} \biggl[ \biggl( p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {} _{\nu}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigl\vert + \biggl( p^{3}q\frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigl\vert \\& \qquad {}+ \biggl( p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert + \biggl( p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert \biggr] . \end{aligned}$$

Thus the proof is completed. □

Corollary 5.4

In Theorem 5.3, for \(p=1\), we derive the new q-trapezoidal inequality

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \biggr\vert \\& \quad \leq \frac{q ( \omega -\nu ) }{8 [ 2 ] _{q} [ 3 ] _{q}} \bigl[ q^{2} \bigl\vert {}_{\nu}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{q}+ [ 2 ] _{q} \bigr) \bigl\vert {}_{\nu} \mathfrak{D}_{q}\hbar ( \omega ) \bigl\vert \\& \qquad {}+ \bigl( [ 3 ] _{q}+ [ 2 ] _{q} \bigr) \bigr\vert {}^{\omega}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert +q^{2} \big\vert {}^{\omega}\mathfrak{D}_{q} \hbar ( \omega ) \big\vert \bigr]. \end{aligned}$$

Example 5.5

Consider \(\hbar : [ 0,1 ] \rightarrow \mathbb{R} \) defined by \(\hbar (x)=x^{3}\). Let also \(q=\frac{1}{3}\) and \(p=\frac{2}{3}\). Then we have the convex functions \({}_{\nu}\mathfrak{D}_{p,q}\hbar ( x ) =\frac{7x^{2}}{27}\) and \({}^{\omega}\mathfrak{D}_{p,q}\hbar ( x ) =\frac{1}{3} ( 7x^{2}+13x+7 ) \). So the left-hand side of (23) can be written as

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad = \biggl\vert \frac{1}{2}-\frac{3}{2} \biggl[ \frac{18}{2520}+ \frac{551}{2520} \biggr] \biggr\vert \\& \quad = \frac{271}{1680}. \end{aligned}$$

On the other hand, since \({}_{\nu}\mathfrak{D}_{p,q}\hbar ( \nu ) =0\), \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \vert =\frac{7}{27}\), \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \vert =\frac{7}{3}\), and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \omega ) \vert =9\), the right-hand side of inequality (23) becomes

$$\begin{aligned}& \frac{q ( \omega -\nu ) }{8 [ 2 ] _{p,q} [ 3 ] _{p,q}} \bigl[ \bigl( [ 3 ] _{p,q}- [ 2 ] _{p,q} \bigr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{p,q}+ [ 2 ] _{p,q} \bigr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigl\vert \\& \qquad {}+ \bigl( [ 3 ] _{p,q}+ [ 2 ] _{p,q} \bigr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{p,q}- [ 2 ] _{p,q} \bigr) \big\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar ( \omega ) \big\vert \bigr] \\& \quad = \frac{3}{56} \biggl[ \biggl( \frac{7}{9}+1 \biggr) \frac{7}{27}+ \biggl( \frac{7}{9}+1 \biggr) \frac{7}{3}+ \biggl( \frac{7}{9}-1 \biggr) 9 \biggr] \\& \quad = \frac{3}{56} \biggl[ \frac{16}{9}\frac{7}{27}+ \frac{16}{9} \frac{7}{3}-2 \biggr] \\& \quad = \frac{3}{56} \biggl[ \frac{16}{9}\frac{7}{27}+ \frac{16}{9} \frac{7}{3}-2 \biggr] \\& \quad = \frac{877}{4536}. \end{aligned}$$

It is clear that

$$ \frac{271}{1680}< \frac{877}{4536}. $$

Theorem 5.6

If Lemma 5.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex for \(s\geq 1\), then

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{q ( \omega -\nu ) }{4p^{2} [ 2 ] _{p,q}} \\& \qquad {} \times \biggl[ \biggl( \biggl( \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{2 [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \biggl( \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{2 [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \biggl( \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{2 [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \nu ) \big\vert ^{s}+ \biggl( \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{2 [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \omega )\big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

Proof

In (24), by the power mean inequality we get

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \int _{0}^{p} ( qt ) \,d_{p,q}t \biggr) ^{1-\frac{1}{s}} \biggl[ \biggl( \int _{0}^{p} ( qt ) \biggl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu +\frac{p+t}{2p} \omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \int _{0}^{p} ( qt ) \bigg\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \bigg\vert ^{s}\,d_{p,q}t \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

By the convexity of the functions \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) we obtain

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{qp^{2}}{ [ 2 ] _{p,q}} \biggr) ^{1-\frac{1}{s}} \\& \qquad {}\times \biggl[ \biggl( \frac{1}{2p} \biggl( p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert ^{s} \\& \qquad {}+\frac{1}{2p} \biggl( p^{3}q \frac{ [ 3 ] _{p,q} + [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {} {}+ \biggl( \frac{1}{2p} \biggl( p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu )\big\vert ^{s} \\& \qquad {}+\frac{1}{2p} \biggl( p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] , \end{aligned}$$

which completes the proof. □

Corollary 5.7

For \(p=1\) in Theorem 5.6, we derive the new q-trapezoidal inequality

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \biggr\vert \\& \quad \leq \frac{q ( \omega -\nu ) }{4 [ 2 ] _{q}} \biggl[ \biggl( \frac{q^{2} \vert {}_{\nu}\mathfrak{D}_{q}\hbar ( \nu ) \vert ^{s}+ ( [ 3 ] _{q}+ [ 2 ] _{q} ) \vert {} _{\nu}\mathfrak{D}_{q}\hbar ( \omega ) \vert ^{s}}{2 [ 3 ] _{q}} \biggr) ^{ \frac{1}{s}} \\& \qquad {}+ \biggl( \frac{ ( [ 3 ] _{q}+ [ 2 ] _{q} ) \vert {}^{\omega}\mathfrak{D}_{q}\hbar ( \nu ) \vert ^{s}+q^{2} \vert {} ^{\omega}\mathfrak{D}_{q}\hbar ( \omega ) \vert ^{s}}{2 [ 3 ] _{q}} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

Theorem 5.8

If Lemma 5.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex for \(s>1\), then

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{ ( qp ) ^{r+1}}{q [ r+1 ] _{p,q}} \biggr) ^{\frac{1}{r}} \\& \qquad {}\times \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {} _{ \nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert ^{s}+ \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}}\big\vert {} ^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr], \end{aligned}$$

where \(s^{-1}+r^{-1}=1\).

Proof

Applying the Hölder inequality to (24), we establish

$$\begin{aligned}& \biggl\vert \frac{\hbar (\nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \int _{0}^{p} ( qt ) ^{r}\,d_{p,q}t \biggr) ^{\frac{1}{r}} \biggl[ \biggl( \int _{0}^{p} \biggl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \\& \qquad {}+ \biggl( \int _{0}^{p} \biggl\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \biggr] . \end{aligned}$$

Since the functions \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex, we have

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{ ( qp ) ^{r+1}}{q [ r+1 ] _{p,q}} \biggr) ^{\frac{1}{r}} \\& \qquad {}\times \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {} _{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{ \nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert ^{s}+ \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar ( \omega )\big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

The proof is ended. □

Corollary 5.9

For \(p=1\) in Theorem 5.8, we derive the new q-trapezoidal inequality

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4} \biggl( \frac{q^{r}}{ [ r+1 ] _{q}} \biggr) ^{\frac{1}{r}} \biggl[ \biggl( \frac{q}{2 ( 1+q ) } \bigl\vert {}_{\nu} \mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{2+q}{2 ( 1+q ) } \bigl\vert {} _{\nu}\mathfrak{D}_{q} \hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{2+q}{2 ( 1+q ) } \big\vert {} ^{ \omega} \mathfrak{D}_{q}\hbar ( \nu ) \big\vert ^{s}+ \frac{q}{2 ( 1+q ) }\big\vert {}^{\omega}\mathfrak{D}_{q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr]. \end{aligned}$$

6 Conclusions

In the present research work, we analyzed a new variant of Hermite–Hadamard inequality in relation to convex functions in the framework of \(( p,q ) \)-calculus. Moreover, we derived some new estimates for \(( p,q ) \)-midpoint and \(( p,q ) \)-trapezoidal inequalities for \(( p,q ) \)-differentiable convex functions using the left and right \(( p,q ) \)-integrals. The upcoming researchers can obtain similar inequalities for different kinds of convexity and coordinated convexity in the context of \(( p,q ) \)-calculus theory in their future research works.

Availability of data and materials

Data sharing not applicable to this paper as no datasets were generated or analyzed during the current study.

References

  1. Kirmaci, U.S.: Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 147(1), 137–146 (2004). https://doi.org/10.1016/S0096-3003(02)00657-4

    Article  MathSciNet  MATH  Google Scholar 

  2. Dragomir, S.S., Agarwal, R.P.: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 11(5), 91–95 (1998). https://doi.org/10.1016/S0893-9659(98)00086-X

    Article  MathSciNet  MATH  Google Scholar 

  3. Dragomir, S.S.: Inequalities of Hermite–Hadamard type for h-convex functions on linear spaces. Proyecciones J. Math. 34(4), 323–341 (2015). https://doi.org/10.4067/S0716-09172015000400002

    Article  MathSciNet  MATH  Google Scholar 

  4. Alomari, M.W., Darus, M., Kirmaci, U.S.: Some inequalities of Hermite–Hadamard type for s-convex functions. Acta Math. Sci. 31(4), 1643–1652 (2011). https://doi.org/10.1016/S0252-9602(11)60350-0

    Article  MathSciNet  MATH  Google Scholar 

  5. Samet, B.: A convexity concept with respect to a pair of functions. Numer. Funct. Anal. Optim. 43(5), 522–540 (2022). https://doi.org/10.1080/01630563.2022.2050753

    Article  MathSciNet  MATH  Google Scholar 

  6. Tunc, M.: On new inequalities for h-convex functions via Riemann–Liouville fractional integration. Filomat 27(4), 559–565 (2013). https://doi.org/10.2298/FIL1304559T

    Article  MathSciNet  MATH  Google Scholar 

  7. Sarikaya, M.Z., Yildrim, H.: On Hermite–Hadamard type inequalities for Riemann–Liouville fractional integrals. Miskolc Math. Notes 17(2), 1049–1059 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Matar, M.M., Abbas, M.I., Alzabut, J., Kaabar, M.K.A., Etemad, S., Rezapour, S.: Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives. Adv. Differ. Equ. 2021, 68 (2021). https://doi.org/10.1186/s13662-021-03228-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Omame, A., Abbas, M., Abdel-Aty, A.: Assessing the impact of SARS-CoV-2 infection on the dynamics of Dengue and HIV via fractional derivatives. Chaos Solitons Fractals 162, 112427 (2022). https://doi.org/10.1016/j.chaos.2022.112427

    Article  MathSciNet  Google Scholar 

  10. Omame, A., Abbas, M., Onyenegecha, C.P.: Backward bifurcation and optimal control in a co-infection model for SARS-CoV-2 and ZIKV. Results Phys. 37, 105481 (2022). https://doi.org/10.1016/j.rinp.2022.105481

    Article  Google Scholar 

  11. Mohammadi, H., Kumar, S., Rezapour, S., Etemad, S.: A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals 144, 110668 (2021). https://doi.org/10.1016/j.chaos.2021.110668

    Article  MathSciNet  Google Scholar 

  12. Etemad, S., Avci, I., Kumar, P., Baleanu, D., Rezapour, S.: Some novel mathematical analysis on the fractal–fractional model of the AH1N1/09 virus and its generalized Caputo-type version. Chaos Solitons Fractals 162, 112511 (2022). https://doi.org/10.1016/j.chaos.2022.112511

    Article  MathSciNet  Google Scholar 

  13. Kumar, P., Erturk, V.S., Abboubakar, H., Nisar, K.S.: Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives. Alex. Eng. J. 60(3), 3189–3204 (2021). https://doi.org/10.1016/j.aej.2021.01.032

    Article  Google Scholar 

  14. Baleanu, D., Etemad, S., Rezapour, S.: On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J. 59(5), 3019–3027 (2020). https://doi.org/10.1016/j.aej.2020.04.053

    Article  Google Scholar 

  15. Nabi, K.N., Abboubakar, H., Kumar, P.: Forecasting of COVID-19 pandemic: from integer derivatives to fractional derivatives. Chaos Solitons Fractals 141, 110283 (2020). https://doi.org/10.1016/j.chaos.2020.110283

    Article  MathSciNet  MATH  Google Scholar 

  16. Rezapour, S., Etemad, S., Mohammadi, H.: A mathematical analysis of a system of Caputo–Fabrizio fractional differential equations for the anthrax disease model in animals. Adv. Differ. Equ. 2020, 481 (2020). https://doi.org/10.1186/s13662-020-02937-x

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, Y., Wang, X.: The evolution of immersed locally convex plane curves driven by anisotropic curvature flow. Adv. Nonlinear Anal. 12(1), 117–131 (2023). https://doi.org/10.1515/anona-2022-0245

    Article  MathSciNet  MATH  Google Scholar 

  18. Eiter, T., Hopf, K., Lasarzik, R.: Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models. Adv. Nonlinear Anal. 12(1), 20220274 (2023). https://doi.org/10.1515/anona-2022-0274

    Article  MathSciNet  MATH  Google Scholar 

  19. Thabet, S.T.M., Etemad, S., Rezapour, S.: On a coupled Caputo conformable system of pantograph problems. Turk. J. Math. 45(1), 496–519 (2021). https://doi.org/10.3906/mat-2010-70

    Article  MathSciNet  MATH  Google Scholar 

  20. Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020, 64 (2020). https://doi.org/10.1186/s13661-020-01361-0

    Article  MathSciNet  MATH  Google Scholar 

  21. Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV-1 infection of \(CD4^{+}\) T-cell with a new approach of fractional derivative. Adv. Differ. Equ. 2020, 71 (2020). https://doi.org/10.1186/s13662-020-02544-w

    Article  MathSciNet  MATH  Google Scholar 

  22. Baleanu, D., Mohammadi, H., Rezapour, S.: Mathematical theoretical study of a particular system of Caputo–Fabrizio fractional differential equations for the Rubella disease model. Adv. Differ. Equ. 2020, 184 (2020). https://doi.org/10.1186/s13662-020-02614-z

    Article  MathSciNet  MATH  Google Scholar 

  23. Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020). https://doi.org/10.1016/j.chaos.2020.109705

    Article  MathSciNet  MATH  Google Scholar 

  24. Rezapour, S., Kumar, S., Iqbal, M.Q., Hussain, A., Etemad, S.: On two abstract Caputo multi-term sequential fractional boundary value problems under the integral conditions. Math. Comput. Simul. 194, 365–382 (2022). https://doi.org/10.1016/j.matcom.2021.11.018

    Article  MathSciNet  MATH  Google Scholar 

  25. Ahmad, B., Nieto, J.J., Alsaedi, A., Al-Hutami, H.: Boundary value problems of nonlinear fractional q-difference (integral) equations with two fractional orders and four-point nonlocal integral boundary conditions. Filomat 28(8), 1719–1736 (2014). https://doi.org/10.2298/FIL1408719A

    Article  MathSciNet  MATH  Google Scholar 

  26. Ren, J., Zhai, C.: A fractional q-difference equation with integral boundary conditions and comparison theorem. Int. J. Nonlinear Sci. Numer. Simul. 18(7–8), 575–583 (2017). https://doi.org/10.1515/ijnsns-2017-0056

    Article  MathSciNet  MATH  Google Scholar 

  27. Rezapour, S., Imran, A., Hussain, A., Martinez, F., Etemad, S., Kaabar, M.K.A.: Condensing functions and approximate endpoint criterion for the existence analysis of quantum integro-difference FBVPs. Symmetry 13(3), 469 (2021). https://doi.org/10.3390/sym13030469

    Article  Google Scholar 

  28. Abdeljawad, T., Alzabut, J.: On Riemann–Liouville fractional q-difference equations and their application to retarded logistic type model. Math. Methods Appl. Sci. 41(18), 8953–8962 (2018). https://doi.org/10.1002/mma.4743

    Article  MathSciNet  MATH  Google Scholar 

  29. Etemad, S., Ntouyas, S.K., Ahmad, B.: Existence theory for a fractional q-integro-difference equation with q-integral boundary conditions of different orders. Mathematics 7(8), 659 (2019). https://doi.org/10.3390/math7080659

    Article  Google Scholar 

  30. Zhang, L., Sun, S.: Existence and uniqueness of solutions for mixed fractional q-difference boundary value problems. Bound. Value Probl. 2019, 100 (2019). https://doi.org/10.1186/s13661-019-1215-z

    Article  MathSciNet  Google Scholar 

  31. Neang, P., Nonlaopon, K., Tariboon, J., Ntouyas, S.K., Ahmad, B.: Existence and uniqueness results for fractional \((p, q)\)-difference equations with separated boundary conditions. Mathematics 10(5), 767 (2022). https://doi.org/10.3390/math10050767

    Article  Google Scholar 

  32. Kamsrisuk, N., Promsakon, C., Ntouyas, S.K., Tariboon, J.: Nonlocal boundary value problems for \((p, q)\)-difference equations. Differ. Equ. Appl. 10(2), 183–195 (2018). https://doi.org/10.7153/dea-2018-10-11

    Article  MathSciNet  MATH  Google Scholar 

  33. Alp, N., Sarikaya, M.Z., Kunt, M., İşcan, I.: q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. J. King Saud Univ., Sci. 30(2), 193–203 (2018). https://doi.org/10.1016/j.jksus.2016.09.007

    Article  MATH  Google Scholar 

  34. Bermudo, S., Kórus, P., Valdés, J.N.: On q-Hermite–Hadamard inequalities for general convex functions. Acta Math. Hung. 162, 364–374 (2020). https://doi.org/10.1007/s10474-020-01025-6

    Article  MathSciNet  MATH  Google Scholar 

  35. Ali, M.A., Budak, M.F.H., Khan, S.: A new version of q-Hermite–Hadamard’s midpoint and trapezoid type inequalities for convex functions. Math. Slovaca (2022, in press)

  36. Sitthiwirattham, T., Ali, M.A., Ali, A., Budak, H.: A new q-Hermite–Hadamard’s inequality and estimates for midpoint type inequalities for convex functions. Miskolc Math. Notes (2022, in press)

  37. Kunt, M., İşcan, I., Alp, N., Sarikaya, M.Z.: \(( p,q ) \)-Hermite–Hadamard inequalities and \(( p,q ) \)-estimates for midpoint inequalities via convex quasi-convex functions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112, 969–992 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Vivas-Cortez, M., Ali, M.A., Budak, H., Kalsoom, H., Agarwal, P.: Some new Hermite–Hadamard and related inequalities for convex functions via \(( p,q ) \)-integral. Entropy 23(7), 828 (2021). https://doi.org/10.3390/e23070828

    Article  MathSciNet  Google Scholar 

  39. Budak, H.: Some trapezoid and midpoint type inequalities for newly defined quantum integrals. Proyecciones 40(1), 199–215 (2021). https://doi.org/10.22199/issn.0717-6279-2021-01-0013

    Article  MathSciNet  MATH  Google Scholar 

  40. Latif, M.A., Kunt, M., Dragomir, S.S., İşcan, I.: Post-quantum trapezoid type inequalities. AIMS Math. 5(4), 4011–4026 (2020). https://doi.org/10.3934/math.2020258

    Article  MathSciNet  MATH  Google Scholar 

  41. Ali, M.A., Budak, H., Zhang, Z., Yildrim, H.: Some new Simpson’s type inequalities for co-ordinated convex functions in quantum calculus. Math. Methods Appl. Sci. 44(6), 4515–4540 (2021). https://doi.org/10.1002/mma.7048

    Article  MathSciNet  MATH  Google Scholar 

  42. Ali, M.A., Abbas, M., Budak, H., Agarwal, P., Murtaza, G., Chu, Y.M.: New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions. Adv. Differ. Equ. 2021, 64 (2021). https://doi.org/10.1186/s13662-021-03226-x

    Article  MathSciNet  MATH  Google Scholar 

  43. Budak, H., Erden, S., Ali, M.A.: Simpson and Newton type inequalities for convex functions via newly defined quantum integrals. Math. Methods Appl. Sci. 44(1), 378–390 (2021). https://doi.org/10.1002/mma.6742

    Article  MathSciNet  MATH  Google Scholar 

  44. Sial, I.B., Mei, S., Ali, M.A., Nonlaopon, K.: On some generalized Simpson’s and Newton’s inequalities for \((\alpha ,m)\)-convex functions in q-calculus. Mathematics 9(24), 3266 (2021). https://doi.org/10.3390/math9243266

    Article  Google Scholar 

  45. Soontharanon, J., Ali, M.A., Budak, H., Nonlaopon, K., Abdullah, Z.: Simpson’s and Newton’s type inequalities for \((\alpha ,m)\)-convex functions via quantum calculus. Symmetry 14(4), 736 (2022). https://doi.org/10.3390/sym14040736

    Article  Google Scholar 

  46. Luangboon, W., Nonlaopon, K., Tariboon, J., Ntouyas, S.K.: Simpson- and Newton-type inequalities for convex functions via \((p,q) \)-calculus. Mathematics 9(12), 1338 (2021). https://doi.org/10.3390/math9121338

    Article  MATH  Google Scholar 

  47. Vivas-Cortez, M.J., Ali, M.A., Qaisar, S., Sial, I.B., Jansem, S., Mateen, A.: On some new Simpson’s formula type inequalities for convex functions in post-quantum calculus. Symmetry 13(12), 2419 (2021). https://doi.org/10.3390/sym13122419

    Article  Google Scholar 

  48. Luangboon, W., Nonlaopon, K., Tariboon, J., Ntouyas, S.K.: On Simpson type inequalities for generalized strongly preinvex functions via \(( p,q) \)-calculus and applications. AIMS Math. 6(9), 9236–9261 (2021). https://doi.org/10.3934/math.2021537

    Article  MathSciNet  MATH  Google Scholar 

  49. Sudsutad, W., Ntouyas, S.K., Tariboon, J.: Quantum integral inequalities for convex functions. J. Math. Inequal. 9(3), 781–793 (2015). https://doi.org/10.7153/jmi-09-64

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhuang, H., Liu, W., Park, J.: Some quantum estimates of Hermite–Hadamard inequalities for quasi-convex functions. Mathematics 7(2), 152 (2019). https://doi.org/10.3390/math7020152

    Article  Google Scholar 

  51. Gauchman, H.: Integral inequalities in q-calculus. Comput. Math. Appl. 47(2–3), 281–300 (2004). https://doi.org/10.1016/S0898-1221(04)90025-9

    Article  MathSciNet  MATH  Google Scholar 

  52. Zhang, Y., Du, T.S., Wang, H., Shen, Y.J.: Different types of quantum integral inequalities via \(( \alpha ,m) \)-convexity. J. Inequal. Appl. 2018, 264 (2018). https://doi.org/10.1186/s13660-018-1860-2

    Article  MathSciNet  MATH  Google Scholar 

  53. Nwaeze, E.R., Tameru, A.M.: New parameterized quantum integral inequalities via η-quasiconvexity. Adv. Differ. Equ. 2019, 425 (2019). https://doi.org/10.1186/s13662-019-2358-z

    Article  MathSciNet  MATH  Google Scholar 

  54. Awan, M.U., Talib, S., Noor, M.A., Noor, K.I., Chu, Y.M.: On post quantum integral inequalities. J. Math. Inequal. 15(2), 629–654 (2021). https://doi.org/10.7153/jmi-2021-15-46

    Article  MathSciNet  MATH  Google Scholar 

  55. Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2001)

    MATH  Google Scholar 

  56. Tariboon, J., Ntouyas, S.K.: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013, 282 (2013). https://doi.org/10.1186/1687-1847-2013-282

    Article  MathSciNet  MATH  Google Scholar 

  57. Tunc, M., Gov, E.: Some integral inequalities via \((p,q)\)-calculus on finite intervals. Filomat 35(5), 1421–1430 (2021). https://doi.org/10.2298/FIL2105421T

    Article  MathSciNet  Google Scholar 

  58. Jackson, F.H.: On a q-definite integrals. Q. J. Pure Appl. Math. 41(15), 193–203 (1910)

    MATH  Google Scholar 

Download references

Acknowledgements

The fourth and fifth authors were supported by Azarbaijan Shahid Madani University. The authors would like to thank the reviewers for their constructive and useful comments. This research has received funding support from the NSRF via the Program Management Unit for Human Resources & lnstitutional Development, Research and lnnovation [grant number B05F640163]. This work is partially supported by the National Natural Science Foundation of China (Grant No. 11971241).

Funding

Not applicable.

Author information

Authors and Affiliations

Authors

Contributions

T.S. and M.A.A. and H.B. and S.E. dealt with the conceptualization, supervision, methodology, investigation, and writing-original draft preparation. T.S. and M.A.A. and H.B. and S.E. and S.R. made the formal analysis, writing-review, editing. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Shahram Rezapour.

Ethics declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sitthiwirattham, T., Ali, M.A., Budak, H. et al. A new version of \(( p,q ) \)-Hermite–Hadamard’s midpoint and trapezoidal inequalities via special operators in \(( p,q ) \)-calculus. Bound Value Probl 2022, 84 (2022). https://doi.org/10.1186/s13661-022-01665-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13661-022-01665-3

MSC

Keywords