Research Article | Open Access
Some New Generalized Integral Inequalities for GA--Convex Functions via Hadamard Fractional Integrals
We prove new generalization of Hadamard, Ostrowski, and Simpson inequalities in the framework of GA--convex functions and Hadamard fractional integral.
Let a real function be defined on a nonempty interval of real line . The function is said to be convex on if inequalityholds for all and
In , Breckner introduced -convex functions as a generalization of convex functions as follows.
Definition 1. Let be a fixed real number. A function is said to be -convex (in the second sense), or that belongs to the class , if for all and .
Of course, -convexity means just convexity when .
The following inequalities are well known in the literature as Hermite-Hadamard inequality, Ostrowski inequality, and Simpson inequality, respectively.
Theorem 2. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds:
Theorem 3. Let be a mapping differentiable in , the interior of I, and let with If , , then the following inequality holds: for all
Theorem 4. Let be a four times’ continuously differentiable mapping on and Then the following inequality holds:
We will give definitions of the right and left hand side Hadamard fractional integrals which are used throughout this paper.
Definition 5. Let . The right-sided and left-sided Hadamard fractional integrals and of order with are defined byrespectively, where is the Gamma function defined by (see ).
Definition 7 (see ). For , a function is said to be GA--convex (geometric-arithmetically -convex) if for all and .
It can be easily seen that if , GA--convexity reduces to GA-convexity.
Lemma 8 (see ). For and , one has where
Let be a differentiable function on , the interior of ; in sequel of this paper we will take where with , , , , and is Euler Gamma function.
In , can gave Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms as follows.
Theorem 9. 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 , can obtained some new inequalities for quasi-geometrically convex functions via fractional integrals by using the following lemma.
Lemma 10. Let be a differentiable function on such that , where with . Then for all , , and one has
In this paper, we will use Lemma 10 to obtain some new inequalities on generalization of Hadamard, Ostrowski, and Simpson type inequalities for GA--convex functions via Hadamard fractional integral.
2. Generalized Integral Inequalities for Some GA--Convex Functions via Fractional Integrals
Theorem 11. Let be a differentiable function on such that , where with . If is GA--convex on in the second sense for some fixed , , , and then the following inequality for fractional integrals holds:where
Proof. Using Lemma 10, property of the modulus, and the power-mean inequality, we have Since is GA--convex on , we getand by a simple computation, we haveHence, If we use (18), (19), and (20) in (17), we obtain the desired result. This completes the proof.
Theorem 19. Let be a differentiable function on such that , where with . If is GA--convex on for some fixed , , , and then the following inequality for fractional integrals holds: where and
Proof. Using Lemma 10, property of the modulus, the Hölder inequality, and GA--convexity of , we have where , andUsing Lemma 8, we have Hence, if we use (32)-(33) in (31) and replacing , we obtain the desired result. This completes the proof.
The authors declare that there are no competing interests regarding the publication of this paper.
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Copyright © 2016 İmdat İşcan and Mustafa Aydin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.