Abstract

The author obtains new estimates on generalization of Hadamard, Ostrowski, and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive real numbers are also given.

1. Introduction

Let real function be defined on some nonempty interval of real line . The function is said to be convex on if inequality holds for all and .

Following are inequalities which are well known in the literature as Hermite-Hadamard inequality, Ostrowski inequality, and Simpson inequality, respectively.

Theorem 1. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds:

Theorem 2. Let be a mapping differentiable in , the interior of I, and let with . If , ; then the following inequality holds: for all .

Theorem 3. Let be four times continuously differentiable mapping on and . Then the following inequality holds:

In recent years, many authors have studied errors estimations for Hermite-Hadamard, Ostrowski, and Simpson inequalities; for refinements, counterparts, and generalization, see [1–9] and references therein.

The following definitions are well known in the literature.

Definition 4. A function is called an -Lipschitzian function on the interval of real numbers with if for all .

For some recent results connected with Hermite-Hadamard type integral inequalities for Lipschitzian functions, see [10–13].

Definition 5 (see [14, 15]). A function is said to be GA-convex (geometric arithmetically convex) if for all and .

We will now give definitions of the right-sided and left-sided Hadamard fractional integrals which are used throughout this paper.

Definition 6. Let . The right-sided and left-sided Hadamard fractional integrals and of oder with are defined by respectively, where is Gamma function defined by (see [16]).

In [17], Iscan established Hermite-Hadamard’s inequalities for GA-convex functions in Hadamard fractional integral forms as follows.

Theorem 7. Let be a function such that , where with . If is a GA-convex function on , then the following inequalities for fractional integrals hold: with .

In the inequality (8), if we take , then we have the following inequality:

Morever, in [17], Iscan obtained a generalization of Hadamard, Ostrowski, and Simpson type inequalities for quasi-geometrically convex functions via Hadamard fractional integrals as related to the inequality (8).

In this paper, the author obtains new general inequalities for Lipschitzian functions via Hadamard fractional integrals as related to the inequality (8).

2. Main Results

Let be a -Lipschitzian function on ; throughout this section, we will take where with , , , , and is Euler Gamma function.

Theorem 8. Let be a -Lipschitzian function on and with . Then for all , , and , we have the following inequality for Hadamard fractional integrals:

Proof. Using the hypothesis of , we have the following inequality:

Corollary 9. In Theorem 8, If we take , then we get In this inequality,(i)if we take , then (ii)If we take , then (iii)If we take and , then

Corollary 10. In Theorem 8, if we take , then we get In this inequality, if we take , then
Specially if we take in this inequality, then we have

Corollary 11. In Theorem 8,(1)if we take and , then  Specially, if we take in this inequality, then we have (2)If we take and , then  Specially, if we take in this inequality, then we have

Corollary 12. In Theorem 8, If we take , then  Specially, if we take in this inequality, then we have We note that if we take , , , and in inequality (25) we obtain inequalities (16), (19), (21), and (23), respectively.

Let be an -Lipschitzian function. In the next theorem, let , , and define , as follows.(1)If , then (2)If , then (3)If , then

Now, we shall give another result for Lipschitzian functions as follows.

Theorem 13. Let and function be defined as above. Then we have the following inequality for Hadamard fractional integrals: