Ostrowski type inequalities via the Katugampola fractional integrals

1.   Introduction

Fractional calculus was first suggested for consideration by Leibnitz in his letter to L'Hospital which dealt with derivatives of order $\alpha = \frac{1}{ 2}$ (see ^[10]). Hereupon, this theory has been used in many fields of science such as economics, biology, engineering, physics and mathematics for sure. Many types of fractional derivatives and integrals were studied by Hadamard, Caputo, Riemann-Liouville, Grünwald-Letnikov, etc. Various properties of these operators have been summarized in ^[9]. For the last decades, this theory has been used in inequality theory frequently because it enables scientists to obtain integral inequalities for also non-integer orders. One of the most famous inequality is Ostrowski's which has lead to gain many practical inequalities with fractional calculus as well.

Ostrowski proved an important integral inequality in 1938 which gives an upper bound for difference between the value $f\left(x\right)$ and mean value of $\ f$ for functions whose derivatives' absolute values are bounded, which can be seen in ^[11] as the following.

Theorem 1. Let $f:I\subseteq \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ and $a, b\in I$ with $a < b.$ If $\left\vert f^{\prime }\left(x\right) \right\vert \leq M$ then,

holds for all $x\in \left[a, b\right]$. Here $\frac{1}{4}$ is the best possible constant.

Zhang and Wan introduced $p-$convex functions in ^[14], and İşcan gave a different version of this definition in ^[5] as follows.

Definition 1. Let $I\subset \left(0, \infty \right)$ be a real interval and $p\in \mathbb{R} \backslash \left\{ 0\right\}$. A function $f:I\rightarrow \mathbb{R}$ is said to be a $p-$convex function, if

for all $x, y$ $\in$ $I$ and $t\in \left[0, 1\right]$.

It is easy to see that $p-$convexity reduces to ordinary convexity for $p = 1$ and harmonically convexity for $p = -1$.

$p-$convex functions are frequently considered in the inequalities especially when using fractional integral calculations. Some fractional integral operators are used to do these calculations. Therefore, some new definitions about fractional calculus are given. First of them is Riemann-Liouville fractional integration operator (see ^[9]) which ables to integrate functions on fractional orders.

Definition 2. Let $f\in L_{1}\left[a, b\right]$. The Riemann-Liouville integrals $J_{a+}^{\alpha }f$ and $J_{b-}^{\alpha }f$ of order $\alpha > 0$ with $a\geqslant 0$ are defined by

and

respectively where $\Gamma \left(\alpha \right) = \int_{0}^{\infty }e^{-t}u^{\alpha -1}du$. Here $J_{a^{+}}^{0}f\left(x\right) = J_{b^{-}}^{0}f\left(x\right) = f\left(x\right)$.

Definition 3. ^[9] The left and right-side Hadamard fractional integrals of order $\alpha \in \mathbb{R} ^{+}$ are defined as

where $\Gamma$ is the gamma function.

Definition 4. ^[8] Let the space $X_{c}^{p}\left(a, b\right)$ ($c\in \mathbb{R}$, $1\leq p\leq \infty$) of those complex-valued Lebesque measurable functions $f$ on $\left[a, b\right]$ for which $\left\Vert f\right\Vert x_{c}^{p} < \infty$, where the norm is defined by

for $1\leq p\leq \infty$, $c\in \mathbb{R}$ and for the case $p = \infty$,

Katugampola revealed a new fractional integration operator which generalizes both Riemann-Liouville and Hadamard fractional integration operators. This integration operator also holds semigroup property (see ^[6,7]) and is defined as the following statement.

Definition 5. Let $\left[a, b\right] \subset \mathbb{R}$ be a finite interval. Then, the left and right-side Katugampola fractional integrals of order $\left(\alpha > 0\right)$ of $f\in X_{c}^{p}\left(a, b\right)$ are defined by

and

with $a < x < b$ and $\rho > 0$ if the integral exists.

Theorem 2. ^[7] Let $\alpha > 0$ and $\rho > 0$. Then for $x > a$,

Similar results also hold for right-sided operators.

Erdélyi et al. deeply involved in hypergeometric functions which Whittaker discovered in 1904 and gave the definition of it in ^[4] as:

Throughout the paper the notation $Y_{f}\left(\alpha, \rho; a, x, b\right)$ will be used in the meaning of following statement.

where $\Gamma$ is Euler Gamma function, i.e., $\Gamma \left(\alpha \right) = \int_{0}^{\infty }e^{-t}u^{\alpha -1}du$.

Alomari et al. proved the following lemma in 2010 in ^[2] to obtain new Ostrowski-type results.

Lemma 1. Let $f:I\subset \mathbb{R} \longrightarrow \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ where $a, b\in I$ with $a < b$. If $f^{\prime }\in L\left[a, b\right]$, then the following equality holds

for each $x\in \left[a, b\right]$.

Set proved the next lemma in 2012 which helps to obtain Ostrowski-type inequalities for Riemann-Liouville fractional integrals in ^[13].

Lemma 2. Let $f:\left[a, b\right] \rightarrow \mathbb{R},$ be a differentiable mapping on $\left(a, b\right)$ with $a < b.$ If $f^{\prime }\in L\left[a, b\right],$ then for all $x\in \left[a, b\right]$ and $\alpha > 0$ the following identity holds

where $\Gamma$ is Euler Gamma function.

To see more studies involving Ostrowski-type inequalities, one can see references ^[1,3,12]. Also in ^[2] and ^[5], Ostrowski-type inequalities using integer order integrals and in ^[13], Ostrowski-type inequalities using Riemann-Liouville integral operator were obtained. On the other hand, the findings in this study were obtained using Katugampola fractional integration operator, which gives more general results than inequalities using integer order integral or Riemann-Liouville fractional integral operator.

In this paper, a new lemma including Katugampola fractional integral has been proved inspired by Lemma 2. Then with the help of some properties and inequalities like Hölder and power mean, new Ostrowski-type inequalities are proved. It is seen that results are supported by the literature.

2.   Main results

Lemma 3. Let $f:I\subset \left(0, \infty \right) \rightarrow \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ where $a, b\in I$ with $a < b.$ If $f^{\prime }\in L\left[a, b\right],$ then for all $x\in \left[a, b\right]$ the following identity holds

where $\alpha > 0$, $\rho > 0$.

Proof. By integrating by parts, the following statement is obtained

With changing the variable $u = \left[tx^{\rho }+\left(1-t\right) a^{\rho } \right] ^{\frac{1}{\rho }}$, it is easy to get

In the same way, integrating by parts $I_{2}$ can be revealed as

With same change of variable, it can be seen that

Multiplying ($2.2$) with $\frac{\left(x^{\rho }-a^{\rho }\right) ^{\alpha +1}}{b-a}$ and ($2.3$) with $\left(-\frac{\left(b^{\rho }-x^{\rho }\right) ^{\alpha +1}}{b-a}\right)$ and summing them side by side, the following calculations can be performed.

With rearranging the last statement

is obtained, which completes the proof.

Remark 1. Under necessary conditions of Lemma $3$ with choosing $\rho = 1$, we get Lemma 2 which is proven in ^[13].

Remark 2. By choosing $\alpha = 1$ in Remark $1$, it is easy to obtain Lemma 1 which is proven in ^[2].

Theorem 3. Let $f:I\subset \left(0, \infty \right) \rightarrow \; \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ and $a, b\in I$ with $a < b$ such that $f^{\prime }\in L\left[a, b\right]$. If $\left\vert f^{\prime }\right\vert$ is $p-$convex on $I$ and $\left\vert f^{\prime }\left(x\right) \right\vert \leq M$ for all $x\in \left[a, 2^{\frac{1}{\rho } }a\right)$ (if $2^{\frac{1}{\rho }}a < b$, otherwise $x\in \left[a, b\right]$), then the following inequality holds

where

and $\rho > 1$, $\alpha > 0$, $\lambda \in \left\{ a, b\right\}$, $_{2}F_{1}\left(., .;.; .\right)$ is hypergeometric function and $Y_{f}\left(\alpha, \rho; a, x, b\right)$ is as defined in (1.4).

Proof. By using Lemma $3$ and properties of modulus, it can be written

By means of $p-$convexity of$\ \left\vert f^{\prime }\right\vert$, following computations can be performed

With necessary computations, it is easy to see that

By using boundedness of $f^{\prime }\left(x\right)$, that is, $\left\vert f^{\prime }\left(x\right) \right\vert \leq M$, it is easy to see

which completes the proof.

Remark 3. By choosing $\rho = 1$ in Theorem 2.1, it reduces to Theorem 7 with $s = 1$ in ^[13] where we used the fact that $_{2}F_{1}\left(x, \text{ }0; \text{ }y; \text{ }z\right) = 1$.

Theorem 4. Let $f:I\subset \left(0, \infty \right) \rightarrow \; \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ and $a, b\in I$ with $a < b$ such that $f^{\prime }\in L\left[a, b\right]$. If $\left\vert f^{\prime }\right\vert ^{q}$ is $p-$convex on $I$ and $\left\vert f^{\prime }\left(x\right) \right\vert \leq M$ for all $x\in I-\left\{ a, b\right\},$ then the following inequality holds

where

and $\rho > 0$, $\alpha > 0$, $\lambda \in \left\{ a, b\right\}$, $r > 1, \frac{1 }{r}+\frac{1}{q} = 1$, $r\neq \frac{\rho }{\rho -1}$ and $Y_{f}\left(\alpha, \rho; a, x, b\right)$ is as defined in (1.4).

Proof. With the help of Lemma $3$ and properties of modulus, one can write

By using Hölder inequality, it can be written as

From the $p-$convexity of $\left\vert f^{\prime }\right\vert ^{q}$ and $\left\vert f^{\prime }\left(x\right) \right\vert \leq M$, it follows that

which completes the proof.

Theorem 5. Let $f:I\subset \left(0, \infty \right) \rightarrow \; \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ and $a, b\in I$ with $a < b$ such that $f^{\prime }\in L\left[a, b\right]$. If $\left\vert f^{\prime }\right\vert ^{q}$ is $p-$convex on $I$ and $\left\vert f^{\prime }\left(x\right) \right\vert \leq M$ for all $x\in \left[a, 2^{\frac{1}{\rho } }a\right)$ (if $2^{\frac{1}{\rho }}a < b$, otherwise $x\in \left[a, b\right]$), then the following inequality holds

where

and $\rho > 1$, $\alpha > 0$, $q > 1$, $\lambda \in \left\{ a, b\right\}$, $_{2}F_{1}\left(., .;.; .\right)$ is hypergeometric function and $Y_{f}\left(\alpha, \rho; a, x, b\right)$ is as defined in (1.4).

Proof. Making use of Lemma 3 and properties of absolute value, it can be seen that

Then, making use of Power-Mean inequality, the following computations can be performed

Hence $\left\vert f^{\prime }\right\vert ^{q}$ is chosen as $p-$convex on $I$

With necessary computations, it can be easily seen that

With using boundedness of $\left\vert f^{\prime }\left(x\right) \right\vert$, it can be written that

So the proof is completed.

Remark 4. By choosing $\rho = 1$ in Theorem 2.3, it reduces to Theorem 9 with $s = 1$ in ^[13] where we used the fact that $_{2}F_{1}\left(x, \text{ }0; \text{ }y; \text{ }z\right) = 1$.

3.   Applications to special means

Let us recall the following means for two positive real numbers.

(1) The arithmetic mean:

(2) The logarithmic mean:

Proposition 1. Let $0 < a < b$ and $\frac{a+b}{2} < 2^{\frac{1}{\rho }}a$. Then the following inequality holds

where $M = \max \left\{ \left\vert \ln a\right\vert, \left\vert \ln b\right\vert \right\}$.

Proof. The proof follows from Theorem 2.1 on applying $\alpha = 1$, $\rho = 2$, $x = \frac{a+b}{2}$ and $f\left(x\right) = -\ln x$ which is $p-$convex on $\left(0, \infty \right)$ for $p\geq 1$.

Proposition 2. Let $0 < a < b$ and $\frac{a+b}{2} < 2^{\frac{1}{\rho }}a$. Then the following inequality holds

Proof. The proof is immediate from Theorem 2.1 on applying $\alpha = 1$, $\rho = 2$, $x = \frac{a+b}{2}$ and $f\left(x\right) = x^{-2}$ which is $p-$ convex on $\left(0, \infty \right)$.

4.   Conclusion

In this study, new lemma and theorems are put forward to obtain new upper bounds for Ostrowski-type inequalities including Katugampola fractional operator. Researchers can obtain new lemmas and theorems by using similar method used in this study or use the obtained results in many fields of science.

Acknowledgments

This research received no external funding. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Conflict of interest

The authors declare no conflict of interest in this paper.