Hermite-Hadamard Type Inequalities Obtained via Fractional Integral for Differentiable -Convex and -Convex Functions

Abstract

Some Hermite-Hadamard type inequalities involving fractional integrals for -convex and ()-convex functions are obtained.

1. Introduction and Preliminaries

The following definition is well known in the literature.

Definition 1. A function , is said to be convex on the interval if the inequality holds for all and .

Geometrically, this means that if , and are three points on the graph of with between and , then is on or below the chord .

Theorem 2 (Hermite-Hadamard inequality). Let be a convex function and with . Then

-convexity was defined by Toader as follows.

Definition 3 (see [1]). The function , is said to be -convex, where , if one has for all and . One says that is -concave if is -convex. Denote by the class of all -convex functions on for which .

Obviously, for , Definition 3 recaptures the concept of standard convex functions on and for the concept of starshaped functions. The notion of -convexity has been further generalized in [2] as it is stated in the following definition.

Definition 4 (see [2]). The function , is said to be ()-convex, where , if one has for all and .
Denote by the class of all ()-convex functions on for which .

It can be easily seen that when one obtains the following classes of functions: convex and -convex, respectively. Note that is a proper subclass of -convex and -functions . The interested reader can find more about partial ordering of convexity in [3].

Definition 5 (see [4]). Let . Then Riemann-Liouville integrals and of order with are defined by where is the Gamma function.

We now give the definition of the hypergeometric series which will be used in obtaining some integrals.

Definition 6 (see [4]). The integral representation of the hypergeometric functions is as follows:where , andis Beta function with

In the present paper, we establish some new Hermite-Hadamard’s type inequalities for the classes of -convex and -convex functions via Riemann-Liouville fractional integrals.

To prove our main results, we consider the following lemma.

Lemma 7 (see [5]). Let be a differentiable function such that . Then, for , , and , one haswhere

2. Generalized Inequalities for -Convex Functions

Theorem 8. Let be on open real interval such that . Let be a differentiable function on such that , and , where . If is an -convex function on for some fixed , thenwhere .

Proof. Using Lemma 7, taking modulus and the fact that is an -convex function, we haveThis completes the proof.

Remark 9. Observe that if in Theorem 8 we have , the statement of Theorem 8 becomes the statement of Theorem   in [6].

Theorem 10. Let be on open real interval such that . Let be a differentiable function on such that , and , where . If is an -convex function on for some fixed and , , thenwhere .

Proof. Using Lemma 7, Hölder’s inequality, and the fact that is an -convex function, This completes the proof.

Remark 11. Observe that if in Theorem 10 we have , the statement of Theorem 10 becomes the statement of Theorem   in [6].

Theorem 12. Let be on open real interval such that . Let be a differentiable function on such that , and , where . If is an -convex function on for some fixed and , thenwhere .

Proof. Using Lemma 7, Power’s mean inequality, and the fact that is an -convex function,This completes the proof.

Remark 13. Observe that if in Theorem 12 we have , the statement of Theorem 12 becomes the statement of Theorem   in [6].

3. Generalized Inequalities for -Convex Functions

Theorem 14. Let be on open real interval such that . Let be a differentiable function on such that , , and , where . If is -convex function on for some fixed , thenwhere and

Proof. Using Lemma 7 and taking modulus and the fact that is -convex function, we haveThis completes the proof.

Remark 15. Observe that if in Theorem 14 we have , the statement of Theorem 14 becomes the statement of Theorem 8.

Theorem 16. Let be on open real interval such that . Let be a differentiable function on such that , , and , where . If is -convex function on for some fixed and , , thenwhere and

Proof. Using Lemma 7, Hölder’s inequality, and the fact that is -convex function, This completes the proof.

Remark 17. Observe that if in Theorem 16 we have , the statement of Theorem 16 becomes the statement of Theorem 10.

Theorem 18. Let be on open real interval such that . Let be a differentiable function on such that , , and , where . If is -convex function on for some fixed and , thenwhere and are given by (19) and .

Proof. Using Lemma 7, Power’s mean inequality, and the fact that is -convex function,This completes the proof.

Remark 19. Observe that if in Theorem 18 we have , the statement of Theorem 18 becomes the statement of Theorem 12.

Competing Interests

The authors declare that they have no competing interests.