Estimates of Upper Bound for a Function Associated with Riemann-Liouville Fractional Integral via -Convex Functions

Abstract

A new identity involving Riemann-Liouville fractional integral is proposed. The result is then used to obtain some estimates of upper bound for a function associated with Riemann-Liouville fractional integral via -convex functions. An application for establishing the inequalities related to special means is also considered.

1. Introduction

A set is said to be convex, if

A function is said to be convex, if

In 1978, Breckner [1] introduced the concept of -convex functions as a generalization of convex functions, as follows.

Definition 1 (see [1]). Let be a real number, . A function is said to be -convex, if

In recent years several new extensions of classical convexity have been proposed in the literature. Varošanec [2] investigated a more generalized class of convex functions named -convex function, as follows.

Definition 2 (see [2]). Let be a nonnegative function. A function is said to be -convex, if

It has been observed that the class of -convex functions unifies several other classes of convexity; for example, if we take , , , and , respectively, then we have the class of -convex functions [1], the class of -functions [3], the class of -Godunova-Levin type functions [4], and the class of -functions [5]. For more details on convexity and its generalizations, see [6–9].

Convexity of a function plays a vital role in theory of inequalities, because many inequalities can easily be obtained using the functions having convexity properties. Hermite and Hadamard’s result which is known as Hermite-Hadamard’s inequality is one of the most fascinating results in the field of integral inequalities. This inequality provides a lower and an upper estimate for the integral average of any convex function defined on an interval. This famous result reads as follows.

Let be a convex function, then

Sarikaya et al. [10] gave a generalization of Hermite-Hadamard’s inequality using the -convexity of the function as follows.

Let be -convex function, then, for , we have

Although the fractional calculus has a long history, it plays significant role in different fields of pure and applied mathematics [11]. Up to now, the study of the fractional calculus is still very active. Sarikaya et al. [12] used the concepts of Riemann-Liouville integrals to obtain the fractional version of Hermite-Hadamard’s inequality. In fact, there are numerous new inequalities that have been obtained using the techniques of fractional calculus. For more details, see [12–15].

In this paper, we present a new integral identity for differentiable functions involving fractional integrals. Then using this auxiliary result we establish our main results that are the estimates of upper bound for a function associated with Riemann-Liouville fractional integral via -convex functions. At the end of the paper, we give an application of the obtained results to the special means.

We begin with recalling the definition of Riemann-Liouville fractional integrals, as follows.

Definition 3 (see [11]). Let . The Riemann-Liouville integrals and of order with are defined by andwhere is the Gamma function.

The integral form of the hypergeometric function is where is the Beta function.

2. Main Results

In this section, we consider the estimates of upper bound for the function below, which is associated with Riemann-Liouville fractional integral. Consider the following:

In order to establish the estimates of upper bound for , we first prove an auxiliary result which plays an important role in dealing with subsequent results.

Lemma 4. Let be differentiable function on with . If , , and , then

Proof. LetIntegrating givesSimilarly integrating , one hasUsing (14) and (15) in (13) leads to the identity described in Lemma 4.

Based on Lemma 4, we are now in a position to establish our main results.

Theorem 5. Let be differentiable function on with . If , , and and is -convex function, then where

Proof. Using Lemma 4 and the fact that is -convex function, we have This completes the proof of Theorem 5.

We now discuss some special cases which can be deduced directly from Theorem 5.

(I) Putting in Theorem 5, we have the following.

Corollary 6. Let be differentiable function on with . If , , and and is convex function, then

(II) Putting in Theorem 5, we have the following.

Corollary 7. Let be differentiable function on with . If , , and and is -convex function, then

(III) Taking in Theorem 5, we have the following.

Corollary 8. Let be differentiable function on with . If , , and and is -function, then

(IV) Taking in Theorem 5, we have the following.

Corollary 9. Let be differentiable function on with . If , , and and is -Godunova-Levin type function, then

Theorem 10. Let be differentiable function on with , , , and , and let be -convex function, . Then

Proof. Using Lemma 4, the Hölder inequality and the fact that is -convex function, we have The proof of Theorem 10 is complete.

We give now four corollaries that follow from the special cases of Theorem 10.

(I) Choosing in Theorem 10, we have the following.

Corollary 11. Let be differentiable function on with , , , and , and let be convex function, . Then

(II) Choosing in Theorem 10, we have the following.

Corollary 12. Let be differentiable function on with , , , and , and let be -convex function, . Then

(III) Putting in Theorem 10, we have the following.

Corollary 13. Let be differentiable function on with , , , and , and let be -function, . Then

(IV) Putting in Theorem 10, we have the following.

Corollary 14. Let be differentiable function on with , , , and , and let be -Godunova-Levin type function, . Then

3. Application to Special Means

In this section, we give an application of the obtained results to special means.

Definition 15 (see [16]). Recall the following definitions.(1) For arbitrary positive numbers (,  is called the logarithmic mean.(2) For arbitrary real numbers ,  is called the arithmetic mean.(3) The extended logarithmic mean of two positive numbers is defined by

We focus on the estimation of upper bound for the difference between logarithmic mean and arithmetic mean; we shall establish two inequalities related to these means.

Proposition 16. If , , then where

Proof. We start by verifying that is -convex on .
In view of which implies that is convex on , thus for all , , and we have Hence, is -convex on
Now, putting , , , in Corollary 7, we obtainandNote that the function is -convex on . Hence, the inequality (32) follows straightway from the inequality given in Corollary 7.

Proposition 17. If , , , , and , thenwhere

Proof. Taking , , in Corollary 12, we obtainand It remains to prove that is -convex on .
In fact, one has It follows that is convex on ; thus for all , , and we have which implies that is -convex on .
Now, utilizing the fact that is -convex on , we can deduce the desired inequality (38) from Corollary 12.

Data Availability

The datasets used or analysed during the current study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

The work of the first author was supported by the Natural Science Foundation of Fujian Province of China (Grant no. 2016J01023) and the Teaching Reform Project of Longyan University (Grant no. 2017JZ02).