Weighted fractional inequalities for new conditions on h-convex functions

Abstract

We use a new function class called B-function to establish a novel version of Hermite–Hadamard inequality for weighted ψ-Hilfer operators. Additionally, we prove two new identities involving weighted ψ-Hilfer operators for differentiable functions. Moreover, by employing these equalities and the properties of the B-function, we derive several trapezoid- and midpoint-type inequalities for h-convex functions. Furthermore, the obtained results are reduced to several well-known and some new inequalities by making specific choices of the function h.

1 Introduction & preliminaries

In recent decades, various publications have focused on generalizing the Hermite–Hadamard inequality and developing trapezoid- and midpoint-type inequalities that provide bounds for the right- and left-hand sides of the aforementioned inequality. The authors [11] demonstrated various similar trapezoid-type inequalities and developed the Hermite–Hadamard inequality for Riemann–Liouville fractional integrals. Kara et al. [8] identified the following Hermite–Hadamard inequalities:

Let $\psi :[a,b]\rightarrow \mathbb{R}$ be a monotone increasing function such that the derivative $\psi ^{\prime }>0$ is continuous on $(a,b)$. If g is a convex function on $[a,b]$, then

$$ \begin{gathered} g \biggl( \frac{a+b}{2} \biggr) \leq \frac{\Gamma (\beta +1)}{2 A_{(\psi ,\beta )}(1)} \biggl[ {^{\beta } \mathcal{J}_{b^{-}}^{\psi }}G \biggl( \frac{a+b}{2} \biggr) +{^{\beta } \mathcal{J}_{a^{+}}^{\psi }}G \biggl( \frac{a+b}{2} \biggr) \biggr] \leq \frac{g ( a ) +g ( b ) }{2},\end{gathered} $$

(1.1)

where the ψ-Hilfer operators are defined as follows:

$$\begin{aligned} &{^{\beta }\mathcal{J}_{a^{+}}^{\psi }}g(x)= \frac{1}{\Gamma (\beta )} \int _{a}^{x}\psi ^{\prime }(t) \bigl(\psi (x)-\psi (t) \bigr)^{\beta -1}g(t) \,d t,\\ &{^{\beta }\mathcal{J}_{b^{-}}^{\psi }}g(x)= \frac{1}{\Gamma (\beta )} \int _{x}^{b}\psi ^{\prime }(t) \bigl(\psi (t)-\psi (x) \bigr)^{\beta -1}g(t) \,d t, \end{aligned}$$

and

$$\begin{aligned} &G(s)=g(s)+g(a+b-s),\\ &A_{(\psi ,\beta )}(1)= \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }+ \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (a) \biggr) ^{\beta }. \end{aligned}$$

See [3, 7, 9, 12] for further information on comparable results.

In [13], the author introduces a novel class of functions, called h-convex functions.

Definition 1

Let $h : J \subseteq \mathbb{R} \rightarrow \mathbb{R}$, where $(0,1) \subseteq J$, be a nonnegative function, $h \neq 0$. We say that $f : I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ is an h-convex function if f is nonnegative and for all $x, y \in I$, $\alpha \in (0, 1)$ we have

$$ f \bigl(\alpha x + (1 - \alpha )y\bigr) \leq h(\alpha )f (x) + h(1 - \alpha )f (y). $$

(1.2)

If the inequality in (1.2) is reversed, then f is said to be h-concave.

By setting

• $h(\lambda )=\lambda$, Definition 1 reduces to that of the classical convex function.

• $h(\lambda )=1$, Definition 1 reduces to that of P-functions [4, 10].

• $h(\lambda )=\lambda ^{s}$, Definition 1 reduces to that of s-convex functions [2].

• $h(\lambda )=\frac{1}{n}\sum_{k=1}^{n}\lambda ^{ \frac{1}{k}}$, Definition 1 reduces to that of polynomial n-fractional convex functions [5].

Recently, the authors of [1] presented a new class of function, called B-function.

Definition 2

Let $a< b$ and $g : (a, b)\subset \mathbb{R} \rightarrow \mathbb{R}$ be a nonnegative function. The function g is a B-function, or g belongs to the class $B(a, b)$, if for all $x\in (a, b)$, we have

$$ g(x-a) +g(b-x)\leq 2 g \biggl(\frac{a+b}{2} \biggr). $$

(1.3)

If the inequality (1.3) is reversed, g is called an A-function, or we say that g belongs to the class $A(a, b)$.

If we have the equality in (1.3), then g is called an AB-function, or we say that g belongs to the class $AB(a, b)$.

Corollary 1

Let $h : (0, 1) \rightarrow \mathbb{R}$ be a nonnegative function. The function h is a B-function if and only if for all $\lambda \in (0, 1)$, we have

$$ h(\lambda ) +h(1-\lambda )\leq 2 h \biggl(\frac{1}{2} \biggr). $$

(1.4)

• The functions $h(\lambda )=\lambda$ and $h(\lambda )=1$ are AB-functions, B-functions, and A-functions.

• The function $h (\lambda )=\lambda ^{s}$, $s\in (0, 1]$ is a B-function.

• The function $h(\lambda ) = \frac{1}{n}\sum _{k=1}^{n}\lambda ^{\frac{1}{k}}, n$, $k\in \mathbb{N}$ is a B-function.

The weighted fractional integrals are defined as follows:

Definition 3

([6])

Let $[a,b]\subseteq {}[ 0,+\infty )$. Let $\beta >0$ and ψ be a positive, increasing differentiable function such that $\psi ^{\prime }(s)\neq 0$ for all $s\in {}[ a,b]$. The left- and right-sided weighted fractional integrals of a function f with respect to the function ψ on $[a,b]$ are respectively defined as follows:

$$\begin{aligned} &{\mathrm{J}_{w, a^{+}}^{\beta ,\psi }}f(x)=\frac{1}{w(x) \Gamma (\beta )} \int _{a}^{x}\psi ^{\prime } ( t ) \bigl( \psi ( x ) -\psi ( t ) \bigr) ^{ \beta -1}w(t)f(t)\,dt,\quad a< x\leq b; \end{aligned}$$

(1.5)

$$\begin{aligned} &{\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}f(x)=\frac{1}{w(x) \Gamma _{k}(\beta )} \int _{x}^{b}\psi ^{\prime } ( t ) \bigl( \psi ( t ) -\psi ( x ) \bigr) ^{\beta -1}w(t)f(t)\,dt,\quad a\leq x< b, \end{aligned}$$

(1.6)

where w is a weighted function and the gamma function defined by

$$ \Gamma (\beta )= \int _{0}^{\infty }t^{\beta -1}e^{-t} \,dt \quad \text{and} \quad \beta \Gamma (\beta )=\Gamma (\beta +1). $$

For these operators, consider the following space:

$$ X[a, b]= \biggl\{ f: \parallel f\parallel _{X}= \biggl( \int _{a}^{b} \bigl\vert w(t) f(t) \bigr\vert \psi ^{\prime }(t)\,dt \biggr)< \infty \biggr\} . $$

For special choices of ψ, w, and β, we get already known results.

• (1) Taking $w(t)=1$, the operators reduce to the ψ-Hilfer integral operators of order $\beta >0$.

• (2) For $\psi (t)=t$, we get the weighted Riemann–Liouville operators.

• (3) For $\psi (t)=t$ and $w(t)=1$, the operators are simplified to Riemann–Liouville integral operators.

• (4) Taking $\psi (t)=t$, $w(t)=1$, and $\beta = 1$, the operators reduce to classical Riemann integrals.

• (5) Setting $\psi (t)= \ln (t)$ and $a>1$, we get the weighted Hadamard operators of order $\beta > 0$.

• (6) Setting $\psi (t)= \ln (t)$, $w(t)=1$, and $a>1$, the operators are simplified to Hadamard operators of order $\beta > 0$.

The purpose of this study is to generalize the Hermite–Hadamard inequality given in [8] for the h-convex function and weighted ψ-Hilfer operator with conditions. For this aim, we assume h is a B-function.

2 Hermite–Hadamard inequality

This section establishes Hermite–Hadamard-type inequalities for h-convex functions using ψ-Hilfer operators. Throughout this paper, we consider that $0\leq a< b<\infty$, $\beta >0$, and ψ is a positive differentiable increasing function on $(a,b)$.

Theorem 2.1

Let h be a B-function and w a nondecreasing function. If $f\in X [a,b]$ is an h-convex function, then the following inequalities hold:

$$ \begin{aligned} \frac{ w(a)}{2 h (\frac{1}{2} )}f \biggl( \frac{a+b}{2} \biggr) & \leq \frac{\Gamma (\beta +1) w(\frac{a+b}{2})}{2 \Omega (\psi ,\beta )} \biggl[{ \mathrm{J} _{w, b^{-}}^{ \beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) + {\mathrm{J} _{w, a^{+}}^{ \beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \\ & \leq 2 h \biggl(\frac{1}{2} \biggr) w(b) \biggl( \frac{f(b)+f(a)}{2} \biggr),\end{aligned}$$

(2.1)

where

$$ F( \tau ) = f(\tau ) + f ( a + b - \tau ) $$

(2.2)

and

$$ \Omega ( \psi , \beta ) = \biggl(\psi (b) - \psi \biggl( \frac{a+b}{2} \biggr) \biggr)^{\beta} + \biggl(\psi \biggl( \frac{a+b}{2} \biggr) - \psi (a) \biggr)^{\beta}. $$

(2.3)

Proof

Since w is a positive nondecreasing function on $[a,b]$,

• (1) for all $\tau \in [a,\frac{a+b}{2}]$, we have $0 < w(a)\leq w(\tau )\leq w(\frac{a+b}{2})\leq w(b)$, and then

  $$\begin{aligned} \begin{aligned} \frac{w(a)}{\beta} \biggl(\psi \biggl( \frac{a+b}{2} \biggr)-\psi ( a) \biggr) ^{\beta} & \leq \int _{a}^{\frac{a+b}{2}} \biggl(\psi \biggl( \frac{a+b}{2} \biggr)-\psi (\tau ) \biggr)^{\beta -1}w(\tau ) \psi '( \tau )\,d\tau \\ & \leq \frac{w(b)}{\beta} \biggl(\psi \biggl( \frac{a+b}{2} \biggr)-\psi (a) \biggr) ^{\beta};\end{aligned} \end{aligned}$$

  (2.4)

• (2) for all $\tau \in [\frac{a+b}{2},b]$, we have $0 < w(a)\leq w(\frac{a+b}{2})\leq w(\tau )\leq w(b)$, and then

  $$ \begin{aligned} \frac{w(a)}{\beta} \biggl(\psi (b)- \psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta} & \leq \int _{\frac{a+b}{2}}^{b} \biggl(\psi (\tau )-\psi \biggl( \frac{a+b}{2} \biggr) \biggr)^{ \beta -1}w(\tau ) \psi '(\tau ) \,d\tau \\ & \leq \frac{w(b)}{\beta} \biggl(\psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta}.\end{aligned}$$

  (2.5)

Letting f be an h-convex function, we have for any $\tau \in {}[ a,b]$,

$$ \begin{aligned} f \biggl( \frac{a+b}{2} \biggr) & =f \biggl( \frac{1}{2}(a+b-\tau )+\frac{1}{2}\tau \biggr) \\ & \leq h \biggl( \frac{1}{2} \biggr) f ( a+b-\tau ) +h \biggl( \frac{1}{2} \biggr) f(\tau ),\end{aligned}$$

and then

$$ f \biggl( \frac{a+b}{2} \biggr) \leq h \biggl( \frac{1}{2} \biggr) F(\tau ). $$

(2.6)

Multiplying (2.6) by $( \psi (\frac{a+b}{2})-\psi (\tau ) ) ^{\beta -1}\psi '( \tau )w(\tau )$ and integrating over $\tau \in {}[ a,\frac{a+b}{2}]$, we obtain

$$ \begin{aligned}& f \biggl( \frac{a+b}{2} \biggr) \int _{a}^{\frac{a+b}{2}}\psi '(\tau ) \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi ( \tau ) \biggr) ^{\beta -1}w(\tau ) \,d\tau \\ &\quad \leq h \biggl( \frac{1}{2} \biggr) \int _{a}^{ \frac{a+b}{2}}\psi '(\tau ) \biggl( \psi \biggl( \frac{a+b}{2} \biggr) - \psi (\tau ) \biggr) ^{\beta -1}w(\tau )F(\tau ) \,d\tau .\end{aligned}$$

By using the left-hand side of (2.4), we deduce

$$ f \biggl( \frac{a+b}{2} \biggr) \biggl( \psi \biggl( \frac{a+b}{2} \biggr)-\psi (a) \biggr) ^{\beta }\leq \frac{h ( \frac{1}{2} ) \Gamma (\beta +1) w(\frac{a+b}{2})}{w(a)}{\mathrm{J} _{w, a^{+}}^{ \beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) . $$

(2.7)

Now, multiplying (2.6) by $( \psi (\tau )-\psi ( \frac{a+b}{2} ) ) ^{ \beta -1}\psi '(\tau )w(\tau )$ and integrating over $\tau \in [ \frac{a+b}{2},b ]$, we get

$$ \begin{aligned} &f \biggl( \frac{a+b}{2} \biggr) \int _{\frac{a+b}{2}}^{b} \biggl( \psi (\tau )-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{ \beta -1}\psi '(\tau )w(\tau ) \,d\tau \\ &\quad \leq h \biggl( \frac{1}{2} \biggr) \int _{ \frac{a+b}{2}}^{b} \biggl( \psi (\tau )-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{ \beta -1}\psi '(\tau )w(\tau )F(\tau ) \,d\tau .\end{aligned}$$

By using the left-hand side of (2.5), we deduce

$$ f \biggl( \frac{a+b}{2} \biggr) \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }\leq \frac{h ( \frac{1}{2} ) \Gamma (\beta +1) w(\frac{a+b}{2})}{w(a)}{ \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) . $$

(2.8)

Adding the inequalities (2.7) and (2.8), we obtain

$$ \begin{gathered} \frac{w(a)}{2 h ( \frac{1}{2} ) }f \biggl( \frac{a+b}{2} \biggr) \leq \frac{\Gamma (\beta +1) w(\frac{a+b}{2})}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta , \psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta , \psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] .\end{gathered} $$

(2.9)

Let us prove the second inequality in (2.1). Since any $\tau \in [ a,b ]$ can be written as $\tau =(1-t)a+tb$ for $t\in [ 0,1 ]$, we have

$$ F(\tau )=f \bigl( (1-t)a+tb \bigr) +f \bigl( ta+(1-t)b \bigr). $$

Applying the h-convexity of the function f, we get

$$ \begin{aligned} F(\tau ) & =f \bigl( (1-t) b+t a \bigr) +f \bigl( (1-t) a+t b \bigr) \\ & \leq h(1-t) \bigl[ f(b)+f(a) \bigr] +h(t) \bigl[ f(b)+f(a) \bigr] \\ & = \bigl( h(t)+h(1-t) \bigr) \bigl[ f(b)+f(a) \bigr] .\end{aligned}$$

Applying (1.4), we deduce

$$ F(\tau )\leq 2 h \biggl( \frac{1}{2} \biggr) \bigl[ f(b)+f(a) \bigr] . $$

(2.10)

Multiplying (2.10) by $( \psi (\frac{a+b}{2})-\psi (\tau ) ) ^{\beta -1}\psi '( \tau )w(\tau )$ and integrating over $\tau \in {}[ a,\frac{a+b}{2}]$, we obtain

$$ \begin{aligned} & \int _{a}^{\frac{a+b}{2}}\psi '(\tau ) \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (\tau ) \biggr) ^{ \beta -1}w(\tau )F(\tau ) \,d\tau \\ &\quad \leq 2 h \biggl( \frac{1}{2} \biggr) \bigl[ f(b)+f(a) \bigr] \int _{a}^{\frac{a+b}{2}}\psi '(\tau ) \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (\tau ) \biggr) ^{\beta -1}w(\tau ) \,d\tau .\end{aligned}$$

By using the right-hand side of (2.4), we deduce

$$ \Gamma (\beta +1){\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \leq \frac{2 h ( \frac{1}{2} ) w(b)}{w(\frac{a+b}{2})} \bigl[ f(b)+f(a) \bigr] \biggl( \psi \biggl(\frac{a+b}{2}\biggr)-\psi (a) \biggr) ^{\beta } .$$

(2.11)

Now, multiplying (2.10) by $( \psi (\tau )-\psi ( \frac{a+b}{2} ) ) ^{ \beta -1}\psi '(\tau )w(\tau )$ and integrating over $\tau \in [ \frac{a+b}{2},b ]$, we get

$$ \begin{aligned} & \int _{\frac{a+b}{2}}^{b} \biggl( \psi ( \tau )-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta -1}\psi '(\tau )w(\tau )F(\tau ) \,d \tau \\ &\quad \leq 2 h \biggl( \frac{1}{2} \biggr) \bigl[ f(b)+f(a) \bigr] \int _{\frac{a+b}{2}}^{b} \biggl( \psi (\tau )-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta -1}\psi '(\tau )w(\tau ) \,d \tau .\end{aligned}$$

By using the right-hand side of (2.5), we deduce

$$ \Gamma (\beta +1){\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \leq \frac{2 h ( \frac{1}{2} ) w(b)}{w(\frac{a+b}{2})} \bigl[ f(b)+f(a) \bigr] \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }. $$

(2.12)

Adding inequalities (2.11) and (2.12), we obtain

$$ \begin{aligned} &\frac{\Gamma (\beta +1) w(\frac{a+b}{2})}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \\ &\quad \leq 2 h \biggl( \frac{1}{2} \biggr) w(b) \biggl( \frac{f(b)+f(a)}{2} \biggr) . \end{aligned} $$

(2.13)

This finishes the proof. □

The following results are dependent on the function h presented in Theorem 2.1. First, assuming $h(\alpha )=\alpha$, we get the following result using the weighted ψ-Hilfer operators for convex functions.

Corollary 2

Let $f\in X [a,b]$ be a convex function. Then the following inequalities hold:

$$ \begin{aligned} w(a) f \biggl( \frac{a+b}{2} \biggr) & \leq \frac{\Gamma (\beta +1) w(\frac{a+b}{2})}{2 \Omega (\psi ,\beta )} \biggl[{\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) + {\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \\ & \leq w(b) \biggl(\frac{f(b)+f(a)}{2} \biggr),\end{aligned} $$

(2.14)

where $F( t)$ and $\Omega (\psi ,\beta )$ are defined by (2.2) and (2.3), respectively.

By setting $h(\alpha )=1$, we get the following result using the weighted ψ-Hilfer operators with an f being a P-function.

Corollary 3

Let $\beta >0$ and $f\in X [a,b]$ be a P-function. Then the following inequalities hold:

$$ \begin{aligned} w(a) f \biggl( \frac{a+b}{2} \biggr) & \leq \frac{\Gamma (\beta +1) w(\frac{a+b}{2})}{\Omega (\psi ,\beta )} \biggl[{ \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) + { \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \\ & \leq 2 w(b) \bigl(f(b)+f(a) \bigr),\end{aligned} $$

(2.15)

where $F( t)$ and $\Omega (\psi ,\beta )$ are defined by (2.2) and (2.3), respectively.

Using $h(\alpha )=\alpha ^{s}$, we obtain the following result through the weighted ψ-Hilfer operators and s-convex functions.

Corollary 4

Let $\beta >0$, $s\in (0, 1]$, and $f\in X [a,b]$ be an s-convex function. Then the following inequalities hold:

$$ \begin{aligned} \frac{ w(a)}{2^{1-s}}f \biggl( \frac{a+b}{2} \biggr) & \leq \frac{\Gamma (\beta +1) w(\frac{a+b}{2})}{2 \Omega (\psi ,\beta )} \biggl[{\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) + {\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \\ & \leq 2^{1-s} w(b) \biggl(\frac{f(b)+f(a)}{2} \biggr).\end{aligned} $$

(2.16)

where $F( t)$ and $\Omega (\psi ,\beta )$ are defined by (2.2) and (2.3), respectively.

Taking $h(\alpha )=\frac{1}{n}\sum_{k=1}^{n}\alpha ^{\frac{1}{k}}$, we deduce the following result through the weighted ψ-Hilfer operators and n-fractional polynomial convex functions.

Corollary 5

Let $\beta >0$ and $f\in X [a,b]$ be an n-fractional polynomial convex function. Then the following inequalities hold:

$$ \begin{aligned} \frac{ w(a)}{C_{n}}f \biggl( \frac{a+b}{2} \biggr) & \leq \frac{\Gamma (\beta +1) w(\frac{a+b}{2})}{2 \Omega (\psi ,\beta )} \biggl[{\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) + {\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \\ & \leq C_{n} w(b) \biggl(\frac{f(b)+f(a)}{2} \biggr).\end{aligned} $$

(2.17)

where $F( t)$, $\Omega (\psi ,\beta )$ are defined by (2.2), (2.3), respectively, and $C_{n}=\frac{2}{n}\sum_{k=1}^{n} (\frac{1}{2} )^{\frac{1}{k}}$.

Remark 1

If we choose $\psi (\tau )=\tau$ and $\psi (\tau )=\ln \tau$ in Corollaries 3, 4, and 5, we obtain Hermite–Hadamard inequality for P-functions, s-convex functions, and n-fractional polynomial convex functions involving the weighted Riemann–Liouville fractional operator and the weighted Hadamard fractional operator, respectively.

3 Weighted trapezoid-type inequalities

This section presents weighted trapezoid inequalities and their particular results utilizing weighted ψ-Hilfer operators with w being symmetric with respect to $\frac{a+b}{2}$ (i.e., $w(t)=w(b+a-t)$). To accomplish this, we must first establish an equality in the following lemma.

Lemma 3.1

Assume w is a differentiable and symmetric with respect to $\frac{a+b}{2}$ function, and suppose h is a B-function. Let $f:[a,b]\rightarrow \mathbb{R}$ be a function where $(wf)$ is a differentiable mapping on $(a,b)$. Then the following identity holds:

$$ \begin{aligned} &\frac{f(a)+f(b)}{2}-\frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \\ &\quad =\frac{b-a}{4 \Phi (\psi ,\beta , w)} \int _{0}^{1}A_{ \psi ,\beta }(\tau )\\ &\qquad \times \biggl[ (wf)^{\prime } \biggl( \frac{1-\tau}{2}a+ \frac{1+\tau}{2}b \biggr) -(wf)^{\prime } \biggl( \frac{1+\tau}{2}a+ \frac{1-\tau}{2}b \biggr) \biggr] \,d\tau ,\end{aligned} $$

(3.1)

where

$$\begin{aligned} \Phi (\psi ,\beta , w) &= \biggl(\psi (b) - \psi \biggl( \frac{a+b}{2} \biggr) \biggr)^{\beta}w(b) + \biggl(\psi \biggl( \frac{a+b}{2} \biggr) - \psi (a) \biggr)^{\beta}w(a). \end{aligned}$$

(3.2)

$$\begin{aligned} A_{\psi , \beta}(\tau ) &= \biggl(\psi \biggl(\frac{a+b}{2} \biggr) - \psi \biggl( \frac{ 1+\tau }{2} a + \frac{ 1-\tau }{2}b \biggr) \biggr)^{\beta} \\ &\quad + \biggl(\psi \biggl( \frac{ 1-\tau }{2} a + \frac{ 1+\tau }{2}b \biggr) - \psi \biggl(\frac{a+b}{2} \biggr) \biggr)^{\beta} . \end{aligned}$$

(3.3)

Proof

Let

$$ J_{1}=\frac{2}{b-a} \int _{a}^{\frac{a+b}{2}} \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (\tau ) \biggr) ^{\beta }(wF)^{\prime }(\tau )\,d\tau . $$

(3.4)

Integrating by parts (3.4) and using (2.2), we get

$$ \begin{aligned} \frac{b-a}{2}J_{1} & = \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (\tau ) \biggr) ^{\beta }w(\tau )F(\tau ) \big|_{a}^{\frac{a+b}{2}} \\ &\quad +\beta \int _{a}^{\frac{a+b}{2}} \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (\tau ) \biggr) ^{\beta -1}\psi ^{\prime }(\tau )w( \tau )F(\tau )\,d\tau .\end{aligned}$$

Therefore

$$ \frac{b-a}{2}J_{1}=- \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (a) \biggr) ^{\beta }w(a)F(a)+\Gamma ( \beta +1) w \biggl( \frac{a+b}{2} \biggr) {\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) . $$

(3.5)

Similarly, let

$$ J_{2}=\frac{2}{b-a} \int _{\frac{a+b}{2}}^{b} \biggl( \psi (\tau )- \psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }(wF)^{\prime }( \tau )\,d\tau . $$

(3.6)

Integrating by parts (3.6), we obtain

$$ \frac{b-a}{2}J_{2}= \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }w(b)F(b)-\Gamma ( \beta +1) w \biggl( \frac{a+b}{2} \biggr) {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) . $$

(3.7)

Since $F(a)=F(b)=f(a)+f(b)$, we conclude from (3.5) and (3.7) that

$$ \begin{aligned} \frac{b-a}{2} ( J_{2}-J_{1} ) &= \Phi (\psi ,\beta ,w) \bigl( f(a)+f(b) \bigr) \\ &\quad -\Gamma (\beta +1) w \biggl( \frac{a+b}{2} \biggr) \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] , \end{aligned} $$

thus

$$ \begin{aligned} &\frac{f(a)+f(b)}{2}-\frac{\Gamma (\beta +1) w ( \frac{a+b}{2} ) }{2 \Phi (\psi ,\beta ,w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \\ &\quad =\frac{b-a}{4 \Phi (\psi ,\beta ,w)} ( J_{2}-J_{1} ) . \end{aligned} $$

(3.8)

On the other hand, since $F^{\prime }(\tau )=f^{\prime }(\tau )-f^{\prime }(a+b-\tau )$ and $w(\tau )=w(a+b-\tau )$, we get

$$ \begin{aligned} (wF)^{\prime }(\tau )&=w^{\prime }(\tau ) \bigl(f(\tau )+f(a+b- \tau )\bigr) + w(\tau ) \bigl(f^{\prime }(\tau )-f^{\prime}(a+b-\tau )\bigr) \\ &=w^{\prime }(\tau )f(\tau )+ w(\tau )f^{\prime }(\tau ) -w^{ \prime }(a+b-\tau )f(a+b-\tau ) \\ &\quad -w(a+b-\tau )f^{\prime}(a+b- \tau )) \\ &= (wf)^{\prime }(\tau )-(wf)^{\prime }(a+b-\tau ). \end{aligned}$$

From (3.4), we get

$$ J_{1}=\frac{2}{b-a} \int _{a}^{\frac{a+b}{2}} \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (\tau ) \biggr) ^{\beta } \bigl( (wf)^{\prime }(\tau )-(wf)^{ \prime }(a+b-\tau ) \bigr) \,d\tau . $$

By changing the variable $\tau =\frac{1+s}{2}a+\frac{1-s}{2}b$, we obtain

$$ \begin{aligned} J_{1}&= \int _{0}^{1} \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr) ^{\beta } \\ &\quad \times \biggl[ (wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) -(wf)^{\prime } \biggl( \frac{1-s}{2}a+\frac{1+s}{2}b \biggr) \biggr] \,ds. \end{aligned} $$

Similarly, from (3.6) we deduce

$$ \begin{aligned} J_{2}&= \int _{0}^{1} \biggl( \psi \biggl( \frac{1-s}{2}a+ \frac{1+s}{2}b \biggr) -\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{ \beta } \\ &\quad \times \biggl[ (wf)^{\prime } \biggl( \frac{1-s}{2}a+ \frac{1+s}{2}b \biggr) -(wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr] \,ds. \end{aligned} $$

Consequently,

$$ J_{2}-J_{1}= \int _{0}^{1}A_{\psi ,\beta }(s) \biggl[ (wf)^{\prime } \biggl( \frac{1-s}{2}a+\frac{1+s}{2}b \biggr) -(wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr] \,ds. $$

(3.9)

Finally, we acquire the needed equality (3.1) by substituting (3.9) into (3.8). □

Remark 2

Putting $w=1$ in Lemma 3.1, we get [8, Lemma 3.1].

Theorem 3.1

Under the hypotheses of Lemma 3.1, if $|(wf)^{\prime }|$ is an h-convex mapping on $[a, b]$ and h is a B-function, then the trapezoid-type inequality holds, namely

$$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a) h(\frac{1}{2})}{2 \Phi (\psi ,\beta , w)} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \,ds.\end{aligned}$$

(3.10)

Proof

Taking the absolute value of the identity (3.1) and using the h-convexity of the function $|(wf)^{\prime }|$, we get

$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w ( \frac{a+b}{2} ) }{2 \Phi (\psi ,\beta ,w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Phi (\psi ,\beta ,w)} \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \biggl[ \biggl\vert (wf)^{\prime } \biggl( \frac{1-s}{2}a+\frac{1+s}{2}b \biggr) \biggr\vert \\ &\qquad + \biggl\vert (wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr\vert \biggr] \,ds \\ &\quad \leq \frac{b-a}{4 \Phi (\psi ,\beta ,w)} \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \biggl[ h \biggl( \frac{1-s}{2} \biggr) +h \biggl( \frac{1+s}{2} \biggr) \biggr] \bigl[ \bigl\vert (wf)^{\prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \,ds,\end{aligned}$$

Given that h is a B-function, setting $\alpha =\frac{1-s}{2}$ and $1-\alpha =\frac{1+s}{2}$ yields

$$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w ( \frac{a+b}{2} ) }{2 \Phi (\psi ,\beta ,w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a) h(\frac{1}{2})}{2 \Phi (\psi ,\beta ,w)}\int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \bigl[ \bigl\vert (wf)^{\prime }(a) \bigr\vert + \bigl\vert (wf)^{ \prime }(b) \bigr\vert \bigr]\,ds.\end{aligned}$$

 □

The following results are obtained via the weighted ψ-Hilfer operators and depend on the function h given in Theorem 3.1.

Corollary 6

• (1) If $|(wf)^{\prime }|$ is a convex mapping on $[a,b]$, then

  $$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w ( \frac{a+b}{2} ) }{2 \Phi (\psi ,\beta ,w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Phi (\psi ,\beta ,w)} \bigl[ \bigl\vert (wf)^{ \prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi , \beta }(s) \bigr\vert \,ds.\end{aligned}$$

  Particularly, putting $w=1$, we get [8, Corollary 3.4].

• (2) If $|(wf)^{\prime }|$ is a P-function on $[a, b]$, then

  $$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{2 \Phi (\psi ,\beta , w)} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \,ds.\end{aligned}$$

• (3) If $|(wf)^{\prime }|$ is an s-convex mapping on $[a, b]$, then

  $$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{2^{s+1} \Phi (\psi ,\beta , w)} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \,ds.\end{aligned}$$

• (4) If $|(wf)^{\prime }|$ is an n-fractional polynomial convex mapping on $[a, b]$, then

  $$ \begin{aligned}& \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a) C_{n}}{4 \Phi (\psi ,\beta , w)} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \,ds,\end{aligned}$$

  where $\Phi (\psi ,\beta , w)$, $A_{\psi ,\beta }(s)$ are defined by (3.2), (3.3), respectively, and $C_{n}=\frac{2}{n}\sum_{k=1}^{n} (\frac{1}{2} )^{\frac{1}{k}}$.

Theorem 3.2

Let $p>1$ and $\frac{1}{p'}+\frac{1}{p}=1$. If $|(wf)^{\prime} |^{p}$ is an h-convex mapping on $[a,b]$, then

$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w (\frac{a+b}{2} )}{2 \Phi (\psi ,\beta , w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a) (2h(\frac{1}{2}))^{\frac{1}{p}}}{4 \Phi (\psi ,\beta , w)} \biggl(2 \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p'}\,ds \biggr) ^{ \frac{1}{p^{\prime }}} \bigl( \bigl\vert (wf)^{\prime}(a) \bigr\vert ^{p}+ \bigl\vert (wf)^{\prime }(b) \bigr\vert ^{p} \bigr) ^{\frac{1}{p}} \\ &\quad \leq \frac{(b-a) (2h(\frac{1}{2}))^{\frac{1}{p}}}{4 \Phi (\psi ,\beta , w)} \biggl(2 \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p'}\,ds \biggr) ^{ \frac{1}{p^{\prime }}} \bigl( \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr).\end{aligned} $$

(3.11)

Proof

Taking absolute value of (3.1) and using the well-known Hölder’s inequality, we obtain

$$ \begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w ( \frac{a+b}{2} ) }{2 \Phi (\psi ,\beta ,w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Phi (\psi ,\beta ,w)} \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \biggl\vert (wf)^{\prime } \biggl( \frac{1-s}{2}a+ \frac{1+s}{2}b \biggr) \biggr\vert \,ds \\ &\qquad +\frac{b-a}{4 \Phi (\psi ,\beta ,w)} \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert \biggl\vert (wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr\vert \,ds \\ &\quad \leq \frac{b-a}{4 \Phi (\psi ,\beta ,w)} \biggl( \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{ \frac{1}{p^{\prime }}}\times \biggl( \int _{0}^{1} \biggl\vert (wf)^{\prime } \biggl( \frac{1-s}{2}a+\frac{1+s}{2}b \biggr) \biggr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}} \\ &\qquad +\frac{b-a}{4 \Phi (\psi ,\beta ,w)} \biggl( \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{ \frac{1}{p^{\prime }}}\times \biggl( \int _{0}^{1} \biggl\vert (wf)^{\prime } \biggl( \frac{1+s}{2}a+\frac{1-s}{2}b \biggr) \biggr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}}.\end{aligned}$$

Notice that for $p>1$, $A,B\geq 0$, $A^{\frac{1}{p}}+B^{\frac{1}{p}}\leq 2^{1-\frac{1}{p}}(A+B)^{\frac{1}{p}}$, and $|(wf)^{\prime } |^{p}$ an h-convex function, we get

$$\begin{aligned} & \biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w ( \frac{a+b}{2} ) }{2 \Phi (\psi ,\beta ,w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Phi (\psi ,\beta ,w)} \biggl( \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{ \frac{1}{p^{\prime }}} 2^{1-\frac{1}{p}} \\ &\qquad \times \biggl[ \int _{0}^{1} \biggl\vert (wf)^{\prime } \biggl( \frac{1-s}{2}a+\frac{1+s}{2} \biggr) \biggr\vert ^{p}\,ds+ \int _{0}^{1} \biggl\vert (wf)^{ \prime } \biggl( \frac{1+s}{2}a+\frac{1-s}{2} \biggr) \biggr\vert ^{p}\,ds \biggr] ^{\frac{1}{p}} \\ &\quad \leq \frac{b-a}{4 \Phi (\psi ,\beta ,w)} \biggl( 2 \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{ \frac{1}{p^{\prime }}} \\ &\qquad \times \biggl( \int _{0}^{1} \biggl[ h \biggl( \frac{1-s}{2} \biggr) +h \biggl( \frac{1+s}{2} \biggr) \biggr] \bigl[ \bigl\vert (wf)^{ \prime }(a) \bigr\vert ^{p}+ \bigl\vert (wf)^{\prime }(b) \bigr\vert ^{p} \bigr]\,ds \biggr) ^{\frac{1}{p}}. \end{aligned}$$

Since h is a B-function, we get

$$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1) w ( \frac{a+b}{2} ) }{2 \Phi (\psi ,\beta ,w)} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{ \mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(b-a) (2h(\frac{1}{2}))^{\frac{1}{p}}}{4 \Phi (\psi ,\beta ,w)} \biggl( 2 \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} \bigl( \bigl\vert (wf)^{ \prime }(a) \bigr\vert ^{p}+ \bigl\vert (wf)^{\prime }(b) \bigr\vert ^{p} \bigr) ^{\frac{1}{p}}.\end{aligned}$$

This proves the first inequality in (3.11).

Notice that the inequality $A^{p}+B^{p}\leq (A+B)^{p}$ yields the second inequality in (3.11). □

Setting $w=1$ and $h(s)=s$ in Theorem 3.2, we get the following corollary.

Corollary 7

Let $p>1$ and $\frac{1}{p^{\prime }}+\frac{1}{p}=1$. If $|f^{\prime} |^{p}$ is a convex mapping on $[a,b]$, then

$$ \begin{aligned} &\biggl\vert \frac{f(a)+f(b)}{2}- \frac{\Gamma (\beta +1)}{2 \Omega (\psi , \beta )} \biggl[ { ^{\beta }\mathcal{J}_{b^{-}}^{ \psi }}F \biggl( \frac{a+b}{2} \biggr) +{ ^{\beta }\mathcal{J}_{a^{+}}^{ \psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Omega (\psi , \beta )} \biggl(2 \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{ \frac{1}{p^{\prime }}} \bigl( \bigl\vert f^{\prime}(a) \bigr\vert ^{p}+ \bigl\vert f^{ \prime }(b) \bigr\vert ^{p} \bigr) ^{\frac{1}{p}} \\ &\quad \leq \frac{b-a}{4 \Omega (\psi , \beta )} \biggl(2 \int _{0}^{1} \bigl\vert A_{\psi ,\beta }(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{ \frac{1}{p^{\prime }}} \bigl( \bigl\vert f^{\prime}(a) \bigr\vert + \bigl\vert f^{ \prime }(b) \bigr\vert \bigr),\end{aligned}$$

(3.12)

which is a better estimate compared with [8, Theorem 3.5].

4 Weighted midpoint-type inequalities

This section establishes some weighted midpoint inequalities for weighted ψ-Hilfer operators using the identity in the following lemma.

Lemma 4.1

Under the hypothesis of Lemma 3.1, the following identity holds:

$$ \begin{aligned} &\frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr]-f \biggl( \frac{a+b}{2} \biggr) \\ &\quad =\frac{b-a}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \\ &\qquad \times \int _{0}^{1} \bigl( \Omega (\psi ,\beta )\\ &\qquad -A_{ \psi ,\beta }(s) \bigr) \biggl[ (wf)^{\prime } \biggl( \frac{1-s}{2}a+ \frac{1+s}{2}b \biggr) -(wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr] \,ds,\end{aligned} $$

(4.1)

where $\Omega (\psi ,\beta )$ and $A_{\psi , \beta}(\tau )$ are defined in (2.3) and (3.3), respectively.

Proof

Let

$$ R_{1}=\frac{2}{b-a} \int _{a}^{\frac{a+b}{2}} \biggl[ \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (a) \biggr) ^{\beta }- \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (\tau ) \biggr) ^{ \beta } \biggr] (wF)^{\prime }(\tau )\,d\tau . $$

(4.2)

By using (3.4), we get

$$\begin{aligned} \frac{b-a}{2}R_{1} & = \int _{a}^{ \frac{a+b}{2}} \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (a) \biggr) ^{ \beta }(wF)^{\prime }( \tau )\,d\tau \\ &\quad - \int _{a}^{\frac{a+b}{2}} \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (\tau ) \biggr) ^{\beta }(wF)^{\prime }( \tau )\,d\tau \\ & = \biggl( \psi \biggl(\frac{a+b}{2}\biggr)-\psi (a) \biggr) ^{ \beta }(wF) (\tau )\big|_{a}^{\frac{a+b}{2}}- \frac{2}{b-a}J_{1} \\ & = \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (a) \biggr) ^{\beta }2(wf) \biggl( \frac{a+b}{2} \biggr) \\ &\quad - \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (a) \biggr) ^{\beta }(wF) (a)-\frac{2}{b-a}J_{1}. \end{aligned}$$

Applying (3.5), we obtain

$$ \begin{aligned} \frac{b-a}{2}R_{1}&=2 \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (a) \biggr) ^{\beta }(wf) \biggl( \frac{a+b}{2} \biggr) \\ &\quad - \Gamma (\beta +1) w \biggl( \frac{a+b}{2} \biggr) {\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) . \end{aligned} $$

(4.3)

Similarly, let

$$ R_{2}=\frac{2}{b-a} \int _{\frac{a+b}{2}}^{b} \biggl[ \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }- \biggl( \psi (\tau )-\psi \biggl(\frac{a+b}{2} \biggr) \biggr) ^{ \beta } \biggr] (wF)^{\prime }(\tau )\,d\tau . $$

(4.4)

Using (3.6), then we have

$$ \begin{aligned} \frac{b-a}{2}R_{2} & = \int _{ \frac{a+b}{2}}^{b} \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{ \beta }(wF)^{\prime }(\tau ) \,d\tau \\ &\quad - \int _{\frac{a+b}{2}}^{b} \biggl( \psi (\tau )-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{ \beta }(wF)^{\prime }(\tau ) \,d\tau \\ & = \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }(wF) (\tau )\big|_{\frac{a+b}{2}}^{b} - \frac{2}{b-a}J_{2} \\ & = \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }(wF) (b)\\ &\quad -2 \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }(wf) \biggl( \frac{a+b}{2} \biggr) - \frac{2}{b-a}J_{2},\end{aligned}$$

and applying (3.7), we get

$$ \begin{aligned} \frac{b-a}{2}R_{2}&=\Gamma (\beta +1) w \biggl( \frac{a+b}{2} \biggr) {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \\ &\quad -2 \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }(wf) \biggl( \frac{a+b}{2} \biggr) . \end{aligned} $$

(4.5)

From (4.3) and (4.5), we have

$$ \begin{aligned} &\frac{b-a}{4 \Omega (\psi ,\beta ) w ( \frac{a+b}{2} ) } ( R_{2}-R_{1} )\\ &\quad = \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta , \psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta , \psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) . \end{aligned} $$

(4.6)

In addition, according to (4.2),

$$ \begin{aligned} R_{1} &=\frac{2}{b-a} \int _{a}^{ \frac{a+b}{2}} \biggl[ \biggl( \psi \biggl( \frac{a+b}{2} \biggr) - \psi (a) \biggr) ^{\beta }- \biggl( \psi \biggl( \frac{a+b}{2} \biggr) - \psi (\tau ) \biggr) ^{\beta } \biggr] \\ &\quad \times \bigl( (wf)^{\prime }(\tau )-(wf)^{ \prime }(a+b-\tau ) \bigr) \,d\tau \\ &= \int _{0}^{1} \biggl[ \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi (a) \biggr) ^{\beta }- \biggl( \psi \biggl( \frac{a+b}{2} \biggr) -\psi \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr) ^{\beta } \biggr] \\ &\quad \times \biggl[ (wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) -(wf)^{\prime } \biggl( \frac{1-s}{2}a+ \frac{1+s}{2}b \biggr) \biggr] \,ds. \end{aligned} $$

Similarly, from (4.4) we get

$$ \begin{aligned} R_{2} &=\frac{2}{b-a} \int _{\frac{a+b}{2}}^{b} \biggl[ \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{ \beta }- \biggl( \psi ( \tau )-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta } \biggr] \\ &\quad \times \bigl( (wf)^{\prime }(\tau )-(wf)^{ \prime }(a+b-\tau ) \bigr) \,d\tau \\ &= \int _{0}^{1} \biggl[ \biggl( \psi (b)-\psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta }- \biggl( \psi \biggl( \frac{1-s}{2}a+ \frac{1+s}{2}b \biggr) - \psi \biggl( \frac{a+b}{2} \biggr) \biggr) ^{\beta } \biggr] \\ &\quad \times \biggl[ (wf)^{\prime } \biggl( \frac{1-s}{2}a+ \frac{1+s}{2}b \biggr) -(wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr] \,ds. \end{aligned} $$

As a result,

$$ \begin{aligned} R_{2}-R_{1}&= \int _{0}^{1} \bigl( \Omega (\psi ,\beta )-A_{\psi , \beta }(s) \bigr) \\ &\quad \times\biggl[ (wf)^{\prime } \biggl( \frac{1-s}{2}a+ \frac{1+s}{2}b \biggr) -(wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2}b \biggr) \biggr] \,ds. \end{aligned} $$

(4.7)

To obtain the desired equality (4.1), substitute (4.7) into (4.6). □

Remark 3

Put $w=1$ in Lemma 4.1, we get [8, Lemma 4.1].

Theorem 4.1

If $|(wf)^{\prime }|$ is an h-convex mapping on $[a, b]$ and h is a B-function, then

$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a) h(\frac{1}{2})}{2 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert \,ds.\end{aligned}$$

(4.8)

Proof

Taking the absolute value of the identity (4.1) and using the h-convexity of $|(wf)^{\prime }|$ and inequality (1.4), we deduce

$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Omega (\psi ,\beta ) w ( \frac{a+b}{2} ) } \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi ,\beta }(s) \bigr\vert \\ &\qquad \times \biggl[ \biggl\vert (wf)^{ \prime } \biggl( \frac{1-s}{2}a+\frac{1+s}{2}b \biggr) \biggr\vert + \biggl\vert (wf)^{\prime } \biggl( \frac{1+s}{2}a+\frac{1-s}{2}b \biggr) \biggr\vert \biggr] \,ds \\ &\quad \leq \frac{b-a}{4 \Omega (\psi ,\beta ) w ( \frac{a+b}{2} ) } \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi ,\beta }(s) \bigr\vert \\ &\qquad \times \biggl[ \biggl( h \biggl( \frac{1-s}{2} \biggr) +h \biggl( \frac{1+s}{2} \biggr) \biggr) \bigl( \bigl\vert (wf)^{\prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr) \biggr] \,ds \\ &\quad =\frac{(b-a) h ( \frac{1}{2} ) }{2 \Omega (\psi ,\beta ) w ( \frac{a+b}{2} ) } \bigl[ \bigl\vert (wf)^{\prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi ,\beta }(s) \bigr\vert \,ds.\end{aligned}$$

This ends the proof. □

The following results are obtained using the weighted ψ-Hilfer operators and depend on the function h given in Theorem 4.1.

Corollary 8

• (1) If $|(wf)^{\prime }|$ is a convex mapping on $[a,b]$, then

  $$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Omega (\psi ,\beta ) w ( \frac{a+b}{2} ) } \bigl[ \bigl\vert (wf)^{\prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi ,\beta }(s) \bigr\vert \,ds.\end{aligned}$$

  Particularly, putting $w=1$, we get [8, Theorem 4.2].

• (2) If $|(wf)^{\prime }|$ is a P-function on $[a, b]$, then

  $$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{2 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert \,ds.\end{aligned}$$

• (3) If $|(wf)^{\prime }|$ is an s-convex mapping on $[a, b]$, then

  $$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{2^{s+1} \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \bigl[ \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr] \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert \,ds.\end{aligned}$$

• (4) If $|(wf)^{\prime }|$ is an n-fractional polynomial convex mapping on $[a,b]$, then

  $$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a) C_{n}}{4 \Omega (\psi ,\beta ) w ( \frac{a+b}{2} ) } \bigl[ \bigl\vert (wf)^{\prime }(a) \bigr\vert + \bigl\vert (wf)^{\prime }(b) \bigr\vert \bigr]\int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi ,\beta }(s) \bigr\vert \,ds,\end{aligned}$$

  where $\Omega (\psi ,\beta )$, $A_{\psi ,\beta }(s)$ are defined by (2.3), (3.3), respectively, and $C_{n}=\frac{2}{n}\sum_{k=1}^{n} ( \frac{1}{2} ) ^{\frac{1}{k}}$.

Theorem 4.2

Let $p>1$ and $\frac{1}{p^{\prime }}+\frac{1}{p}=1$. If $|(wf)^{\prime} |^{p}$ is an h-convex mapping on $[a,b]$, then

$$\begin{aligned} \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ & \quad \leq \frac{(b-a) (2h(\frac{1}{2}))^{\frac{1}{p}}}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \biggl(2 \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} \\ &\qquad \times\bigl( \bigl\vert (wf)^{\prime}(a) \bigr\vert ^{p}+ \bigl\vert (wf)^{ \prime }(b) \bigr\vert ^{p} \bigr) ^{\frac{1}{p}} \\ &\quad \leq \frac{(b-a) (2h(\frac{1}{2}))^{\frac{1}{p}}}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \biggl(2 \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} \bigl( \bigl\vert (wf)^{\prime}(a) \bigr\vert + \bigl\vert (wf)^{ \prime }(b) \bigr\vert \bigr).\end{aligned} \end{aligned}$$

(4.9)

Proof

Taking the absolute value of (4.1) and using the well-known Hölder’s inequality, we obtain

$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert \biggl\vert (wf)^{\prime } \biggl( \frac{1-s}{2}a+\frac{1+s}{2}b \biggr) \biggr\vert \,ds \\ &\qquad +\frac{b-a}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert \biggl\vert (wf)^{\prime } \biggl( \frac{1+s}{2}a+\frac{1-s}{2}b \biggr) \biggr\vert \,ds \\ &\quad \leq \frac{b-a}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \biggl( \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}}\\ &\qquad \times\biggl( \int _{0}^{1} \biggl\vert (wf)^{\prime } \biggl( \frac{1-s}{2}a+ \frac{1+s}{2} \biggr) \biggr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}} \\ &\qquad +\frac{b-a}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \biggl( \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} \\ &\qquad \times\biggl( \int _{0}^{1} \biggl\vert (wf)^{\prime } \biggl( \frac{1+s}{2}a+ \frac{1-s}{2} \biggr) \biggr\vert ^{p}\,ds \biggr) ^{\frac{1}{p}}.\end{aligned}$$

Noticing that $A^{\frac{1}{p}}+B^{\frac{1}{p}}\leq 2^{1-\frac{1}{p}}(A+B)^{\frac{1}{p}}$ and $|(wf)^{\prime } |^{p}$ is an h-convex function, we conclude

$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{b-a}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \biggl( \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} 2 ^{1-\frac{1}{p}} \\ &\qquad \times \biggl[ \int _{0}^{1} \biggl\vert (wf)^{\prime } \biggl( \frac{1-s}{2}a+\frac{1+s}{2} \biggr) \biggr\vert ^{p}\,ds+ \int _{0}^{1} \biggl\vert (wf)^{ \prime } \biggl( \frac{1+s}{2}a+\frac{1-s}{2} \biggr) \biggr\vert ^{p}\,ds \biggr] ^{\frac{1}{p}} \\ &\quad \leq \frac{b-a}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \biggl(2 \int _{0}^{1} \bigl\vert \Omega (\psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} \\ &\qquad \times \biggl( \int _{0}^{1} \biggl[h \biggl( \frac{1-s}{2} \biggr)+ h \biggl(\frac{1+s}{2} \biggr) \biggr] \bigl[ \bigl\vert (wf)^{ \prime}(a) \bigr\vert ^{p}+ \bigl\vert (wf)^{\prime }(b) \bigr\vert ^{p} \bigr]\,ds \biggr) ^{\frac{1}{p}}.\end{aligned}$$

Putting $\alpha =\frac{1-s}{2}$ and $1-\alpha =\frac{1+s}{2}$ yields

$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ { \mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ &\quad \leq \frac{(b-a) (2h(\frac{1}{2}))^{\frac{1}{p}}}{4 \Omega (\psi ,\beta ) w (\frac{a+b}{2} )} \biggl(2 \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} \bigl( \bigl\vert (wf)^{\prime}(a) \bigr\vert ^{p}+ \bigl\vert (wf)^{ \prime }(b) \bigr\vert ^{p} \bigr) ^{\frac{1}{p}}.\end{aligned}$$

This proves the first inequality in (4.9).

The second inequality in (4.9) is clear from the inequality $A^{p}+B^{p}\leq (A+B)^{p}$. □

Setting $w=1$ and $h(s)=s$ in Theorem 4.2, we get the following corollary.

Corollary 9

Let $p>1$ and $\frac{1}{p^{\prime }}+\frac{1}{p}=1$. If $|f^{\prime} |^{p}$ is a convex mapping on $[a,b]$, then

$$ \begin{aligned} & \biggl\vert \frac{\Gamma (\beta +1)}{2 \Omega (\psi ,\beta )} \biggl[ {\mathrm{J} _{w, b^{-}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) +{\mathrm{J} _{w, a^{+}}^{\beta ,\psi }}F \biggl( \frac{a+b}{2} \biggr) \biggr] -f \biggl( \frac{a+b}{2} \biggr) \biggr\vert \\ & \quad \leq \frac{b-a}{4 \Omega (\psi , \beta )} \biggl(2 \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} \bigl( \bigl\vert f^{\prime}(a) \bigr\vert ^{p}+ \bigl\vert f^{ \prime }(b) \bigr\vert ^{p} \bigr) ^{\frac{1}{p}} \\ & \quad \leq \frac{b-a}{4 \Omega (\psi , \beta )} \biggl(2 \int _{0}^{1} \bigl\vert \Omega ( \psi ,\beta )-A_{\psi , \beta}(s) \bigr\vert ^{p^{\prime }}\,ds \biggr) ^{\frac{1}{p^{\prime }}} \bigl( \bigl\vert f^{\prime}(a) \bigr\vert + \bigl\vert f^{ \prime }(b) \bigr\vert \bigr),\end{aligned}$$

(4.10)

which is a better estimate compared with [8, Theorem 4.5].

5 Conclusions

In this study, we recalled a new function class, namely that of B-functions, and utilized it to derive a novel version of the Hermite–Hadamard inequality for weighted ψ-Hilfer operators. We also established two new identities involving weighted ψ-Hilfer operators for differentiable functions. By combining these identities and the properties of the B-function, we obtained several trapezoid- and midpoint-type inequalities for h-convex functions. Our results not only extend the existing literature on inequalities involving fractional operators but also provide new insights into the behavior of h-convex functions under these operators. Additionally, our methods can be applied to other fractional integral operators by using B-functions.