Research Article | Open Access
İmdat İşcan, Mustafa Aydin, "Some New Generalized Integral Inequalities for GA--Convex Functions via Hadamard Fractional Integrals", Chinese Journal of Mathematics, vol. 2016, Article ID 4361806, 8 pages, 2016. https://doi.org/10.1155/2016/4361806
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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