Extension of Milne-type inequalities to Katugampola fractional integrals

Abstract

This study explores the extension of Milne-type inequalities to the realm of Katugampola fractional integrals, aiming to broaden the analytical tools available in fractional calculus. By introducing a novel integral identity, we establish a series of Milne-type inequalities for functions possessing extended s-convex first-order derivatives. Subsequently, we present an illustrative example complete with graphical representations to validate our theoretical findings. The paper concludes with practical applications of these inequalities, demonstrating their potential impact across various fields of mathematical and applied sciences.

1 Introduction

Definition 1.1

([24])

A function $\mathfrak{g}:I\rightarrow \mathbb{R}$ is said to be convex if

$$ \mathfrak{g}\left ( tu+\left ( 1-t\right ) v\right ) \leq t \mathfrak{g}\left ( u\right ) +\left (1-t\right ) \mathfrak{g}(v) $$

holds for all $u,v\in I$ and all $t\in \lbrack 0,1]$.

The notion of convexity is a fundamental concept in both pure and applied mathematics, offering a rich framework for understanding the behavior of functions, sets, and optimization problems. It plays a pivotal role in the theory of approximation, especially in integral inequalities. This is because convex functions preserve inequalities under integration, a property that is crucial for establishing bounds and estimates in various approximation techniques. This concept has been extended to include variations like s-convexity [7], P-convexity [16], Godunova-Levin convexity [18], and s-Godunova-Levin convexity [15], broadening the scope to include a wider array of functions. These extensions allow for greater flexibility in mathematical and applied research, accommodating functions that do not meet the strict criteria of classic convexity yet still retain useful convex-like properties.

Definition 1.2

([16])

A nonnegative function $\mathfrak{g}:I\rightarrow \mathbb{R}$ is said to be P-convex if

$$ \mathfrak{g}\left ( tu+\left ( 1-t\right ) v\right ) \leq \mathfrak{g}\left ( u\right ) +\mathfrak{g}\left (v\right ) $$

holds for all $u,v\in I$ and all $t\in \lbrack 0,1]$.

Definition 1.3

([18])

A nonnegative function $\mathfrak{g}:I\rightarrow \mathbb{R}$ is said to be a Godunova-Levin function if

$$ \mathfrak{g}\left ( tu+\left ( 1-t\right ) v\right ) \leq \tfrac{\mathfrak{g}\left ( u\right ) }{t}+\tfrac{\mathfrak{g}(v)}{1-t} $$

holds for all $u,v\in I$ and all $t\in \left ( 0,1\right )$.

Definition 1.4

([15])

A nonnegative function $\mathfrak{g}:I\rightarrow \mathbb{R}$ is said to be an s-Godunova-Levin function, where $s\in \left [ 0,1\right ]$, if

$$ \mathfrak{g}\left ( tu+\left ( 1-t\right ) v\right ) \leq \tfrac{\mathfrak{g}\left ( u\right ) }{t^{s}}+\tfrac{\mathfrak{g}(v)}{\left ( 1-t\right ) ^{s}} $$

holds for all $u,v\in I$ and all $t\in \left ( 0,1\right )$.

Definition 1.5

([7])

A nonnegative function $\mathfrak{g}:I\subset \left [ 0,\infty \right ) \rightarrow \mathbb{R}$ is said to be s-convex in the second sense for some fixed $s\in \left ( 0,1\right ]$ if

$$ \mathfrak{g}\left (tu+(1-t)v\right )\ \leq t^{s}\mathfrak{g}\left (u \right )+(1-t)^{s}\mathfrak{g}\left (v\right ) $$

holds for all $u,v\in I$ and $t\in \lbrack 0,1]$.

Definition 1.6

([32])

A nonnegative function $\mathfrak{g}:I\subset \left [ 0,\infty \right ) \rightarrow \mathbb{R}$ is said to be extended s-convex for some fixed $s\in \left [ -1,1\right ]$ if

$$ \mathfrak{g}\left (tu+(1-t)v\right )\ \leq t^{s}\mathfrak{g}\left (u \right )+(1-t)^{s}\mathfrak{g}\left (v\right ) $$

holds for all $u,v\in I$ and $t\in \left ( 0,1\right )$.

Numerous researchers have delved into utilizing these varied types of convexity to establish error bounds for a range of quadrature rules [4, 15, 16, 25, 26, 32].

Fractional calculus, an extension of traditional calculus, delves into the concepts of derivatives and integrals of non-integer order, offering a broader framework for analyzing dynamical systems. Several fractional integral operators were introduced, such as the Riemann-Liouville [22], Hadamard [28], and Katugampola fractional operators [19], among others, each providing distinct approaches to defining and computing fractional derivatives and integrals. These operators expand the mathematical tool kit, allowing for more nuanced descriptions of memory and hereditary properties in various physical and engineering processes.

Among the most renowned quadrature rules is the Milne rule, a powerful numerical method for approximating the integral of a function and that originates from the broader family of Simpson’s rule. The accuracy of Milne’s rule has been extensively studied through various works that explore error estimation associated with this formula across different types of integrals and classes of functions: via the classical Riemann integral [3], local fractional integral [2, 23], quantum integral [5, 31], the Riemann-Liouville fractional operators [8–10], the proportional Caputo-Hybrid operator [12], and the conformable fractional operator [33]. For further studies on this formula, we refer readers to [6, 13, 14, 27].

In [14], the authors established the following Milne-type inequality for functions whose derivatives are convex as follows:

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( a\right ) - \mathfrak{g}\left ( \tfrac{a+b}{2}\right ) +2\mathfrak{g}\left ( b\right ) \right ) -\tfrac{1 }{ b-a} \underset{a}{\overset{b}{\int}} \mathfrak{g}(z)\ dz \right \vert \\ \leq &\tfrac{5\left (b-a\right )}{24}\left (\left \vert \mathfrak{g}^{ \prime }\left ( a\right ) \right \vert +\left \vert \mathfrak{g}^{ \prime }\left (b\right ) \right \vert \right ) . \end{aligned}$$

Budak et al. [10] extended the above-mentioned result to Riemann-Liouville fractional integrals:

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( a\right ) - \mathfrak{g}\left ( \tfrac{a+b}{2}\right ) +2\mathfrak{g}\left ( b\right ) \right ) - \tfrac{2^{\alpha -1}\Gamma \left ( \alpha +1\right ) }{\left ( b-a\right ) ^{\alpha }} \left (J_{a^{+}}^{\alpha }\mathfrak{g}\left ( \tfrac{a+b}{2}\right )+J_{b^{-}}^{ \alpha }\mathfrak{g}\left ( \tfrac{a+b}{2}\right )\right ) \right \vert \\ \leq &\tfrac{b-a}{12}\left (\tfrac{\alpha +4 }{ \alpha +1}\right ) \left (\left \vert \mathfrak{g}^{\prime }\left ( a\right ) \right \vert +\left \vert \mathfrak{g}^{\prime }\left (b\right ) \right \vert \right ), \end{aligned}$$

where $\mathfrak{g}$ is a convex function, and $J_{a^{+}}^{\alpha}$ and $J_{b^{-}}^{\alpha}$ represent the left- and right-sided Riemann-Liouville fractional integrals, respectively.

Among the different fractional integral operators that have been introduced, the Katugampola fractional integral operators are designed to provide a more general framework that encompasses both the Riemann-Liouville and Hadamard fractional integrals. Introduced by Udita N. Katugampola in 2011, this approach offers unified fractional integral operators with a parameter that can adjust the definition to match the specific requirements of various applications. Since the introduction of these operators, several works on integral inequalities using these operators have been published. In [17], Farid et al. established Ostrowski-type inequalities for differentiable h-convex functions. Kermausuor provided some generalized Ostrowski-type inequalities for functions with strongly $(s, m)$-convex second-order derivatives in [20] and Simpson-type inequalities for functions with s-convex first-order derivatives in [21]. For additional studies in this direction, we direct readers to [1, 11, 29, 30] and the references cited therein.

In this work, we extend Milne-type inequalities to Katugampola fractional integrals. To this end, we first introduce a new integral identity on which we base the establishment of several Milne-type inequalities for functions with extended s-convex first-order derivatives. Subsequently, we present an example that includes graphical representations to confirm the validity of our results. Finally, we conclude the study with some applications, demonstrating the practical implications and potential of our findings in broader mathematical and applied contexts.

2 Preliminaries

In this section, we present key definitions that are crucial to further conducting our research.

Definition 2.1

([22])

Let $\mathfrak{g}\in L_{1}[a,b]$. The fractional Riemann-Liouville integrals $J_{a^{+}}^{\alpha }f$ and $J_{b^{-}}^{\alpha }f$ of order $\alpha >0$ are defined by

$$\begin{aligned} J_{a^{+}}^{\alpha }\mathfrak{g}(x) =& \frac{1}{\Gamma \left ( \alpha \right ) } \overset{x}{\underset{a}{\int }}\left ( x-v\right ) ^{\alpha -1}\mathfrak{g}(v)dv, \text{ \ \ \ }x>a, \\ J_{b^{-}}^{\alpha }\mathfrak{g}(x) =& \frac{1}{\Gamma \left ( \alpha \right ) } \overset{b}{\underset{x}{\int }}\left ( v-x\right ) ^{\alpha -1}\mathfrak{g}(v)dv, \text{ \ \ \ }b>x, \end{aligned}$$

respectively, where $\Gamma (\mu )=\overset{\infty }{\underset{0}{\int }}$ $e^{-v}v^{\mu -1}dv$ is the Gamma function, and $J_{a^{+}}^{0}\mathfrak{g}(x)=J_{b^{-}}^{0}\mathfrak{g}(x)= \mathfrak{g}(x)$.

Definition 2.2

([28])

Let $\alpha >0$. The left- and right-sided Hadamard’s fractional integral operators of order $\alpha >0$ are defined by

$$\begin{aligned} H_{a^{+}}^{\alpha }\mathfrak{g}(x) =& \frac{1}{\Gamma \left ( \alpha \right ) } \overset{x}{\underset{a}{\int }}\left ( \ln x-\ln v\right ) ^{\alpha -1} \frac{\mathfrak{g}(v)}{v}dv,\text{ \ \ \ }x>a, \\ H_{b^{-}}^{\alpha }\mathfrak{g}(x) =& \frac{1}{\Gamma \left ( \alpha \right ) } \overset{b}{\underset{x}{\int }}\left ( \ln v-\ln x\right ) ^{\alpha -1} \frac{\mathfrak{g}(v)}{v}dv,\text{ \ \ \ }b>x. \end{aligned}$$

In what follows, $X_{k}^{r}\left ( a,b\right )$ $\left ( k\in \mathbb{R}\right.$ and $\left .1\leq r\leq \infty \right )$ denotes the space of all complex-valued Lebesgue measurable functions $\mathfrak{g}$ for which $\left \Vert \mathfrak{g}\right \Vert _{X_{k}^{r}}<\infty$, where

$$ \left \Vert \mathfrak{g}\right \Vert _{X_{k}^{r}}=\left ( \underset{a}{\overset{b}{\int}}\left \vert v^{k}\mathfrak{g}(v) \right \vert ^{r}\right )^{\frac{1}{r}} \ \text{for}\ 1\leq r< \infty , $$

and for $r=\infty$

$$ \left \Vert \mathfrak{g}\right \Vert _{X_{k}^{\infty}}=ess \underset{a\leq v\leq b}{\sup}\left \vert v^{k}\mathfrak{g}(v)\right \vert . $$

Definition 2.3

([19])

Let $\mathfrak{g}\in X_{k}^{r}\left ( a,b\right )$. The left- and right-sided Katugampola fractional integrals of order $\alpha \in \mathbb{C}$ with $Re\left ( \alpha \right ) >0$ and $\rho >0$ are defined by

The Katugampola fractional integral operators are crafted to offer a broader framework that includes the Riemann-Liouville and Hadamard fractional integrals, as elucidated in the following remark:

Remark 2.4

For $\alpha , \rho >0$ and $x>a$, we have

• 1. ,

• 2. .

Equivalent results also apply to right-sided operators.

Definition 2.5

([22])

For any complex numbers and nonpositive integers x, y such that $Re\left ( x\right ) >0$ and $Re\left ( y\right ) >0$, the beta function is defined by

$$ B\left ( x,y\right ) =\overset{1}{\underset{0}{\int }}v^{x-1}\left ( 1-v \right ) ^{y-1}dv= \tfrac{\Gamma \left ( x\right ) \Gamma \left ( y\right ) }{\Gamma \left ( x+y\right ) }, $$

where $\Gamma \left ( .\right )$ is the Euler gamma function.

Definition 2.6

([22])

The hypergeometric function is defined as follows

where $c>b>0$, $\left \vert z\right \vert <1$, and $B\left ( .,.\right )$ is the beta function.

3 Main results

The following lemma is necessary to justify our results.

Lemma 3.1

Let $\mathfrak{g}:\left [ a^{\rho },b^{\rho }\right ] \rightarrow \mathbb{R}$ be a differentiable function on $\left [ a^{\rho },b^{\rho }\right ]$ with $0\leq a< b$, and $\mathfrak{g}^{\prime }\in L^{1}\left [ a^{\rho },b^{\rho }\right ]$. Then the following equality holds

where $\alpha ,\rho >0$ and

Proof

Let

$$ I_{1}=\underset{0}{\overset{1}{\int }}\left ( t^{\rho \alpha }+ \tfrac{1}{3}\right ) t^{\rho -1}\mathfrak{g}\left ( \left ( 1-t^{\rho }\right ) \tfrac{a^{\rho }+b^{\rho }}{2}+t^{\rho }b^{\rho }\right ) dt $$

and

$$ I_{2}=\underset{0}{\overset{1}{\int }}\left ( t^{\rho \alpha }+ \tfrac{1}{3}\right ) t^{\rho -1}\mathfrak{g}^{\prime }\left ( t^{\rho }a^{\rho }+ \left ( 1-t^{\rho }\right ) \tfrac{a^{\rho }+b^{\rho }}{2}\right ) dt. $$

Integrating by parts $I_{1}$, we get

Similarly, we have

Subtract (3.4) from (3.3), then multiplying the resulting equality by $\tfrac{\rho \left ( b^{\rho }-a^{\rho }\right ) }{4}$, we get the desired result. □

Theorem 3.2

Let $\mathfrak{g}:\left [ a^{\rho },b^{\rho }\right ] \rightarrow \mathbb{R}$ be a differentiable function on $\left [ a^{\rho },b^{\rho }\right ]$ with $0\leq a< b$, and $\mathfrak{g}^{\prime }\in L^{1}\left [ a^{\rho },b^{\rho }\right ]$. If $\left \vert \mathfrak{g}^{\prime }\right \vert$ is extended s-convex in the second sense for some fixed $s\in \left ( -1,1\right ]$, then we have

where $\alpha ,\rho >0$, is defined as in (3.2), and $B\left ( .,.\right )$ is the beta function.

Proof

Using the properties of modulus and the extended s-convexity of $\left \vert \mathfrak{g}^{\prime }\right \vert$, from Lemma 3.1, we have

where we have used the fact that

$$ \underset{0}{\overset{1}{\int }}\left ( t^{\rho \alpha }+\tfrac{1}{3} \right ) t^{\rho -1}t^{s\rho }dt= \tfrac{4\left ( s+1\right ) +\alpha }{3\rho \left ( s+1\right ) \left ( s+1+\alpha \right ) } $$

(3.5)

and

$$ \underset{0}{\overset{1}{\int }}\left ( t^{\rho \alpha }+\tfrac{1}{3} \right ) t^{\rho -1}\left ( 1-t^{\rho }\right ) ^{s}dt= \tfrac{1}{\rho }B\left ( \alpha +1,s+1\right ) + \tfrac{1}{3\rho \left ( s+1\right ) }. $$

(3.6)

The proof is completed. □

Corollary 3.3

In Theorem 3.2, if $\left \vert \mathfrak{g}^{\prime }\right \vert$ is a P-function, then we obtain

Corollary 3.4

In Theorem 3.2, if $\left \vert \mathfrak{g}^{\prime }\right \vert$ is a convex function, then we obtain

Corollary 3.5

In Theorem 3.2, setting $\rho =1$ yields

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( a\right ) - \mathfrak{g}\left ( \tfrac{a+b}{2}\right ) +2\mathfrak{g}\left ( b\right ) \right ) - \tfrac{2^{\alpha -1}\Gamma \left ( \alpha +1\right ) }{\left ( b-a\right ) ^{\alpha }} \mathcal{J}^{\alpha }\left ( \mathfrak{g}\right ) \right \vert \\ \leq &\tfrac{b-a}{4}\left ( \tfrac{4\left ( s+1\right ) +\alpha }{3\left ( s+1\right ) \left ( s+1+\alpha \right ) }\left ( \left \vert \mathfrak{g}^{\prime }\left ( a\right ) \right \vert +\left \vert \mathfrak{g}^{\prime }\left ( b\right ) \right \vert \right ) \right . \\ &+\left . 2\left ( B\left ( \alpha +1,s+1\right ) + \tfrac{1}{3\left ( s+1\right ) }\right ) \left \vert \mathfrak{g}^{\prime }\left ( \tfrac{a+b}{2} \right ) \right \vert \right ) , \end{aligned}$$

where

$$ \mathcal{J}^{\alpha }\left ( \mathfrak{g}\right ) =J_{a^{+}}^{\alpha } \mathfrak{g}\left ( \tfrac{a+b}{2}\right ) +J_{b^{-}}\mathfrak{g}\left ( \tfrac{a+b}{2}\right ) . $$

(3.7)

Corollary 3.6

In Corollary 3.5, setting $\alpha =1$ yields

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( a\right ) - \mathfrak{g}\left ( \tfrac{a+b}{2}\right ) +2\mathfrak{g}\left ( b\right ) \right ) -\tfrac{1}{b-a} \underset{a}{\overset{b}{\int }}\mathfrak{g}\left ( z\right ) dz\right \vert \\ \leq &\tfrac{b-a}{4\left ( s+1\right ) \left ( s+2\right ) }\left ( \tfrac{4s+5}{3}\left \vert \mathfrak{g}^{\prime }\left ( a\right ) \right \vert + \tfrac{2s+10}{3}\left \vert \mathfrak{g}^{\prime }\left ( \tfrac{a+b}{2}\right ) \right \vert +\tfrac{4s+5}{3}\left \vert \mathfrak{g}^{\prime }\left ( b\right ) \right \vert \right ) , \end{aligned}$$

which, for $s \in (0,1]$, corresponds to Theorem 2.2 in [14].

Corollary 3.7

In Corollary 3.5, if we take $s=0$, we get

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( a\right ) - \mathfrak{g}\left ( \tfrac{a+b}{2}\right ) +2\mathfrak{g}\left ( b\right ) \right ) - \tfrac{2^{\alpha -1}\Gamma \left ( \alpha +1\right ) }{\left ( b-a\right ) ^{\alpha }} \mathcal{J}^{\alpha }\left ( \mathfrak{g}\right ) \right \vert \\ \leq &\tfrac{b-a}{4\left ( 1+\alpha \right ) }\left ( \tfrac{4+\alpha }{3}\left \vert \mathfrak{g}^{\prime }\left ( a\right ) \right \vert + \tfrac{8+2\alpha }{3}\left \vert \mathfrak{g}^{\prime }\left ( \tfrac{a+b}{2}\right ) \right \vert +\tfrac{4+\alpha }{3}\left \vert \mathfrak{g}^{\prime }\left ( b\right ) \right \vert \right ) . \end{aligned}$$

Moreover, if we take $\alpha =1$, we obtain

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( a\right ) - \mathfrak{g}\left ( \tfrac{a+b}{2}\right ) +2\mathfrak{g}\left ( b\right ) \right ) -\tfrac{1}{b-a} \underset{a}{\overset{b}{\int }}\mathfrak{g}\left ( z\right ) dz\right \vert \\ \leq &\tfrac{5\left ( b-a\right ) }{24}\left ( \left \vert \mathfrak{g}^{\prime }\left ( a\right ) \right \vert +2\left \vert \mathfrak{g}^{\prime }\left ( \tfrac{a+b}{2}\right ) \right \vert + \left \vert \mathfrak{g}^{\prime }\left ( b\right ) \right \vert \right ) . \end{aligned}$$

Corollary 3.8

In Corollary 3.5, if we take $s=1$, we get

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( a\right ) - \mathfrak{g}\left ( \tfrac{a+b}{2}\right ) +2\mathfrak{g}\left ( b\right ) \right ) - \tfrac{2^{\alpha -1}\Gamma \left ( \alpha +1\right ) }{\left ( b-a\right ) ^{\alpha }} \mathcal{J}^{\alpha }\left ( \mathfrak{g}\right ) \right \vert \\ \leq &\tfrac{b-a}{4}\left ( \tfrac{8+\alpha }{6\left ( 2+\alpha \right ) }\left \vert \mathfrak{g}^{\prime }\left ( a\right ) \right \vert + \tfrac{6+\left ( 1+\alpha \right ) \left ( 2+\alpha \right ) }{3\left ( 1+\alpha \right ) \left ( 2+\alpha \right ) } \left \vert \mathfrak{g}^{\prime }\left ( \tfrac{a+b}{2}\right ) \right \vert +\tfrac{8+\alpha }{6\left ( 2+\alpha \right ) }\left \vert \mathfrak{g}^{\prime }\left ( b\right ) \right \vert \right ) . \end{aligned}$$

This result represents a refinement of that obtained by Budak et al. in Theorem 1 [10] since the latter can be deduced using the convexity of $\left \vert f'\right \vert$, i.e., $\left \vert f'\left (\frac{a+b}{2}\right )\right \vert \leq \frac{1}{2}\left (\left \vert f'(a)\right \vert + \left \vert f'(b) \right \vert \right )$.

Remark 3.9

If we set $\alpha =1$ in Corollary 3.8, we recover the same result established in Corollary 2.4 [14].

Theorem 3.10

Let $\mathfrak{g}:\left [ a^{\rho },b^{\rho }\right ] \rightarrow \mathbb{R}$ be a differentiable function on $\left [ a^{\rho },b^{\rho }\right ]$ with $0\leq a< b$, and $\mathfrak{g}^{\prime }\in L^{1}\left [ a^{\rho },b^{\rho }\right ]$. If $\left \vert \mathfrak{g}^{\prime }\right \vert ^{\kappa }$ is extended s-convex in the second sense for some fixed $s\in \left ( -1,1\right ]$, where $\kappa >1$ with $\frac{1}{\lambda }+\frac{1}{\kappa }=1$, then we have

where is defined as in (3.2), and $_{2}F_{1}\left ( .,.,.;.\right )$ is the hypergeometric function.

Proof

From Lemma 3.1, we have

Using the Hölder inequality and the extended s-convexity in the second sense of $\left \vert \mathfrak{g}^{\prime }\right \vert ^{\kappa }$, we get

where we have used the fact that

The proof is completed. □

Corollary 3.11

In Theorem 3.10, if $\left \vert \mathfrak{g}^{\prime }\right \vert ^{\kappa }$ is a P-function, then we obtain

Corollary 3.12

In Theorem 3.10, if $\left \vert \mathfrak{g}^{\prime }\right \vert ^{\kappa }$ is a convex function, then we obtain

Theorem 3.13

Let $\mathfrak{g}:\left [ a^{\rho },b^{\rho }\right ] \rightarrow \mathbb{R}$ be a differentiable function on $\left [ a^{\rho },b^{\rho }\right ]$ with $0\leq a< b$, and $\mathfrak{g}^{\prime }\in L^{1}\left [ a^{\rho },b^{\rho }\right ]$. If $\left \vert \mathfrak{g}^{\prime }\right \vert ^{\kappa }$ is extended s-convex in the second sense for some fixed $s\in \left ( -1,1\right ]$, where $\kappa \geq 1$ with $\frac{1}{\lambda }+\frac{1}{\kappa }=1$, then we have

where is defined as in (3.2), and $B\left ( .,.\right )$ is the beta function.

Proof

Using power mean inequality in (3.8), since $\left \vert \mathfrak{g}^{\prime }\right \vert ^{\kappa }$ is extended s-convex in the second sense, we have

The proof is achieved. □

Corollary 3.14

In Theorem 3.13, if $\left \vert \mathfrak{g}^{\prime }\right \vert ^{\kappa }$ is a P-function, then we obtain

Corollary 3.15

In Theorem 3.13, if $\left \vert \mathfrak{g}^{\prime }\right \vert ^{\kappa }$ is a convex function, then we obtain

4 Validation and applications

This section is dedicated to validating the accuracy of the findings presented in this study through a numerical example with graphical representations, as well as illustrating the practical application and implications of these results.

4.1 Graphical validation

In this subsection, our goal is to confirm the accuracy of our results through an example supported by graphs generated in Matlab. These visual representations are intended to directly illustrate the precision of our findings.

Example 4.1

Consider the interval $[a, b] = [0, 1]$ and let $\mathfrak{g}: [0, 1] \rightarrow \mathbb{R}$ be a real function defined by $\mathfrak{g}(z) =\tfrac{z^{s+1}}{s+1}$, where the derivative $\mathfrak{g}'(z) =z^{s}$ represents an s-convex function for $s\in (0,1]$.

From Theorem 3.2, we derive the following value for the left- and right-hand sides, as depicted in Fig. 1:

and

$$ RHS=\tfrac{1}{4}\left [ \tfrac{4\left ( s+1\right ) +\alpha }{3\left ( s+1\right ) \left ( s+1+\alpha \right ) }+ 2^{1-s} \left ( B\left ( \alpha +1,s+1\right ) + \tfrac{1}{3\left ( s+1\right ) }\right ) \right ]. $$

From the representations in Fig. 1, we observe that the right-hand side consistently lies above the left-hand side, confirming the accuracy of our results.

Figure 1

The diagrammatic presentation to Theorem 3.2 for $s\in (0,1]$ and $\alpha \in (0,2]$

4.2 Application to quadrature formulas

Let ϒ denote the segmentation of the interval $\left [a, b\right ]$ at the points $a = z_{0} < z_{1} < \cdots < z_{n} = b$. Consider the following numerical integration formula:

$$ \int _{a}^{b} \mathfrak{g}(z) dz = \lambda (\mathfrak{g}, \Upsilon ) + R(\mathfrak{g}, \Upsilon ), $$

where

$$ \lambda (\mathfrak{g}, \Upsilon ) = \sum _{i=0}^{n-1} \tfrac{z_{i+1} - z_{i}}{3} \left (2\mathfrak{g}(z_{i}) - \mathfrak{g} \left (\tfrac{z_{i} + z_{i+1}}{2}\right ) + 2\mathfrak{g}(z_{i+1}) \right ) $$

represents the quadrature’s approximation, and $R(\mathfrak{g}, \Upsilon )$ signifies the error in approximation associated with this method.

Proposition 1

Let $n\in \mathbb{N}$ and $\mathfrak{g}:\left [ a,b\right ] \rightarrow \mathbb{R}$ be a differentiable function on $\left [ a,b\right ]$ with $0\leq a< b$ and $\mathfrak{g}^{\prime }\in L^{1}\left [ a,b\right ]$. If $\left \vert \mathfrak{g}^{\prime }\right \vert$ is a P-function, we have

$$ \left \vert R\left ( f,\Upsilon \right ) \right \vert \leq \underset{i=0}{\overset{n-1}{\sum }}\tfrac{5\left ( z_{i+1}-z_{i}\right ) ^{2}}{24} \left ( \left \vert \mathfrak{g}^{\prime }\left ( z_{i}\right ) \right \vert +2\left \vert \mathfrak{g}^{\prime }\left ( \tfrac{z_{i}+z_{i+1}}{2}\right ) \right \vert +\left \vert \mathfrak{g}^{\prime }\left ( z_{i+1}\right ) \right \vert \right ) . $$

Proof

Applying second inequality of Corollary 3.7 on the subintervals $\left [ z_{i},z_{i+1}\right ]$ $( i=0,1,\ldots, n-1 )$ of the partition ϒ, we get

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( z_{i}\right ) - \mathfrak{g}\left ( \tfrac{z_{i}+z_{i+1}}{2}\right ) +2\mathfrak{g}\left ( z_{i+1}\right ) \right ) -\tfrac{1}{z_{i+1}-z_{i}}\underset{z_{i}}{\overset{z_{i+1}}{\int }}\mathfrak{g} \left ( z\right )\ dz\right \vert \\ \leq &\tfrac{5\left ( z_{i+1}-z_{i}\right ) }{24}\left ( \left \vert \mathfrak{g}^{\prime }\left ( z_{i}\right ) \right \vert +2\left \vert \mathfrak{g}^{\prime }\left ( \tfrac{z_{i}+z_{i+1}}{2}\right ) \right \vert +\left \vert \mathfrak{g}^{ \prime }\left ( z_{i+1}\right ) \right \vert \right ) . \end{aligned}$$

Multiplying both sides of the above inequality by $\left ( z_{i+1}-z_{i}\right )$, then summing the obtained inequalities for $i=0,1,\ldots,n-1$ and using the triangular inequality, we get the desired result. □

Proposition 2

Let $n\in \mathbb{N}$ and $\mathfrak{g}:\left [ a,b\right ] \rightarrow \mathbb{R}$ be a differentiable function on $\left [ a,b\right ]$ with $0\leq a< b$ and $\mathfrak{g}^{\prime }\in L^{1}\left [ a,b\right ]$. If $\left \vert \mathfrak{g}^{\prime }\right \vert$ is convex, we have

$$ \left \vert R\left ( f,\Upsilon \right ) \right \vert \leq \underset{i=0}{\overset{n-1}{\sum }}\tfrac{\left ( z_{i+1}-z_{i}\right ) ^{2}}{24} \left ( 3\left \vert \mathfrak{g}^{\prime }\left ( z_{i}\right ) \right \vert +4\left \vert \mathfrak{g}^{\prime }\left ( \tfrac{z_{i}+z_{i+1}}{2}\right ) \right \vert +3\left \vert \mathfrak{g}^{\prime }\left ( z_{i+1}\right ) \right \vert \right ) . $$

Proof

Applying Corollary 3.8 with $\alpha =1$ on the subintervals $\left [ z_{i},z_{i+1}\right ]$ $\left ( i=0,1,\ldots,n-1\right )$ of the partition ϒ, we get

$$\begin{aligned} &\left \vert \tfrac{1}{3}\left ( 2\mathfrak{g}\left ( z_{i}\right ) - \mathfrak{g}\left ( \tfrac{z_{i}+z_{i+1}}{2}\right ) +2\mathfrak{g}\left ( z_{i+1}\right ) \right ) -\tfrac{1}{z_{i+1}-z_{i}}\underset{z_{i}}{\overset{z_{i+1}}{\int }}\mathfrak{g} \left ( z\right ) \ dz\right \vert \\ \leq &\tfrac{z_{i+1}-z_{i}}{24}\left ( 3\left \vert \mathfrak{g}^{ \prime }\left ( z_{i}\right ) \right \vert +4\left \vert \mathfrak{g}^{ \prime }\left ( \tfrac{z_{i}+z_{i+1}}{2}\right ) \right \vert +3\left \vert \mathfrak{g}^{\prime }\left ( z_{i+1} \right ) \right \vert \right ) . \end{aligned}$$

Multiplying both sides of the above inequality by $\left ( z_{i+1}-z_{i}\right )$, then summing the obtained inequalities for $i=0,1,\ldots,n-1$ and using the triangular inequality, we get the desired result. □

4.3 Application to special means

For any given real numbers a and b, the following definitions apply:

The arithmetic mean is given by $A(a, b) = \tfrac{a + b}{2}$.

The p-logarithmic mean is defined as $L_{p}(a, b) = \left ( \tfrac{b^{p+1} - a^{p+1}}{(p+1)(b - a)} \right )^{\frac{1}{p}}$, where $a, b > 0$, $a \neq b$, and $p \in \mathbb{R} \setminus \{-1, 0\}$.

Proposition 3

Let $a,b\in \mathbb{R}$ and $\kappa ,\lambda >1$, with $0< a< b$ and $\frac{1}{\kappa }+\frac{1}{\lambda }=1$, then we have

$$\begin{aligned} &\left \vert 4A\left ( a^{2},b^{2}\right ) -A^{2}\left ( a,b\right ) -3L_{2}^{2} \left ( a,b\right ) \right \vert \\ \leq &\tfrac{b-a}{2}\left ( \tfrac{4^{\lambda +1}-1}{3}\right ) ^{ \frac{1}{\lambda }}\left ( \left ( a^{\kappa }+\left ( \tfrac{a+b}{2}\right ) ^{ \kappa }\right ) ^{\frac{1}{\kappa }}+\left ( \left ( \tfrac{a+b}{2} \right ) ^{\kappa }+b^{\kappa }\right ) ^{\frac{1}{\kappa }}\right ) . \end{aligned}$$

Proof

The statement is derived from Corollary 3.11 by setting both ρ and α equal to 1 and applying this to the function $\mathfrak{g}(z) = \frac{1}{2}z^{2}$. □

5 Conclusion

In conclusion, this work successfully extends Milne-type inequalities to incorporate Katugampola fractional integrals, marking a significant advancement in the field of fractional calculus. By developing a new integral identity, we have laid the groundwork for analyzing functions with extended s-convex first-order derivatives more comprehensively. The example provided confirms the correctness of our theoretical results, supported by graphical evidence. Furthermore, the applications discussed illustrate the broad utility of our findings, potentially benefiting a wide range of scientific disciplines. Our study not only enriches the existing literature on integral inequalities but also encourages further exploration and application of fractional calculus techniques in addressing intricate mathematical challenges.