Hermite-Hadamard-Fejer type inequalities for GA-s convex functions via fractional integrals

Hermite-Hadamard-Fejer type inequalities for GA-s convex functions via fractional integrals Imdat Iscan1 and Mehmet Kunt 2 1Department of Mathematics, Faculty of Sciences and Arts, Gi resun University, Giresun, Turkey 2Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, Trabzon, Turkey Received: 4 January 2016, Revised: 4 January 2016, Accepted : 1 May 2016 Published online: 26 May 2016.


Introduction
Let f : I R ! R be a convex function defined on the interval I of real numbers and a; b 2 I with a\b. The inequality is well known in the literature as Hermite-Hadamard's inequality (Hadamard 1893; Hermite 1883). The most well-known inequalities related to the integral mean of a convex function f are the Hermite-Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejér inequalities. Fejér (1906) established the following Fejér inequality which is the weighted generalization of Hermite-Hadamard inequality (1): Theorem 1 Let f : a; b ½ ! R be convex function. Then the inequality holds, where w : a; b ½ ! R is nonnegative, integrable and symmetric to ða þ bÞ=2: For some results which generalize, improve, and extend the inequalities (1) and (2), see Bombardelli and Varošanec (2009), Chen and Wu (2014), Dragomir and Agarwal (1998), Fang and Shi (2014), İşcan (2013, 2014c, d, 2016b, c), Mihai et al. (2015), Noor et al. (2016), Pearce and Pecaric (2000), Sarıkaya (2012) and Tseng et al. (2011).
We will now give definitions of the right-hand side and left-hand side Riemann-Liouville fractional integrals which are used throughout this paper.
Definition 1 (Kilbas et al. 2006). Let f 2 L a; b ½ . The right-hand side and left-hand side Riemann-Liouville fractional integrals J a aþ f and J a bÀ f of order a [ 0 with b [ a ! 0 are defined by J a aþ f ðxÞ ¼ respectively, where CðaÞ is the Gamma function defined by CðaÞ ¼ R 1 0 e Àt t aÀ1 dt. Because of the wide application of Hermite-Hadamard type inequalities and fractional integrals, many researchers extend their studies to Hermite-Hadamard type inequalities involving fractional integrals not limited to integer integrals. Recently, more and more Hermite-Hadamard inequalities involving fractional integrals have been obtained for different classes of functions; see Dahmani (2010), İşcan (2014a, b, 2015), İşcan and Wu (2014), İşcan et al. (2016d), Sarıkaya et al. (2013) and Wang et al. (2012Wang et al. ( , 2013. İşcan (2014d) gave the definition of harmonically convex function and established the following Hermite-Hadamard type inequality for harmonically convex functions as follows: for all x; y 2 I and t 2 0; 1 ½ . If the inequality in (3) is reversed, then f is said to be harmonically concave.
Theorem 2 (İşcan 2014d). Let f : I & Rn 0 f g ! R be a harmonically convex function and a; b 2 I with a\b. If f 2 L a; b ½ ; then the following inequalities hold: Chen and Wu (2014) presented Hermite-Hadamard-Fejér inequality for harmonically convex functions as follows: Theorem 3 Let f : I Rn 0 f g ! R be a harmonically convex function and a; b 2 I with a\b. If f 2 L a; b ½ and w : a; b ½ Rn 0 f g ! R is nonnegative, integrable and harmonically symmetric with respect to 2ab aþb , then Sarıkaya et al. (2013) presented Hermite-Hadamard inequality for convex functions via fractional integrals as follows: Theorem 4 Let f : a; b ½ ! R be a positive function with 0 a\b and f 2 L a; b ½ . If f is a convex function on a; b ½ , then the following inequalities for fractional integrals hold: ð6Þ with a [ 0. İşcan and Wu (2014) presented Hermite-Hadamard inequality for harmonically convex functions via fractional integrals as follows: Theorem 5 Let f : I 0; 1 ð Þ!R be a function such that f 2 L a; b ½ , where a; b 2 I with a\b. If f is a harmonically convex function on a; b ½ , then the following inequalities for fractional integrals hold: İşcan (2015) presented Hermite-Hadamard-Fejér inequality for convex functions via fractional integrals as follows: Theorem 6 Let f : a; b ½ ! R be a convex function with a\b and f 2 L a; b ½ . If w is nonnegative, integrable and symmetric to a þ b ð Þ=2, then the following inequalities for fractional integrals hold: with a [ 0. İşcan et al. (2016d) presented Hermite-Hadamard-Fejér inequality for harmonically convex functions via fractional integrals as follows: Theorem 7 Let f : a; b ½ ! R be a harmonically convex function with a\b and f 2 L a; b ½ . If w : a; b ½ ! R is nonnegative, integrable and harmonically symmetric with respect to 2ab=a þ b, then the following inequalities for fractional integrals holds: with a [ 0 and gðxÞ ¼ 1 Zhang and Wan (2007) gave the definition of p-convex function on I & R, İşcan (2016c) gave a different definition of p-convex function on I & 0; 1 ð Þ as follows: for all x; y 2 I and t 2 0; 1 ½ .
It can be easily seen that for p ¼ 1 and p ¼ À1, pconvexity reduces to ordinary convexity and harmonically convexity of functions defined on I & 0; 1 ð Þ, respectively. In Fang and Shi (2014), Theorem 5, if we take I & 0; 1 ð Þ, p 2 Rn 0 f g and h t ð Þ ¼ t, then we have the following theorem.
Theorem 8 Let f : I & 0; 1 ð Þ!R be a p-convex function, p 2 Rn 0 f g, and a; b 2 I with a\b. If f 2 L a; b ½ then the following inequalities hold: For some results related to p-convex functions and its generalizations, we refer the reader to see Fang and Shi (2014), İşcan (2016aİşcan ( , b, c), Mihai et al. (2015, Noor et al. (2016) and Zhang and Wan (2007).
In this paper, we built Hermite-Hadamard-Fejér type inequalities for p-convex functions in fractional integral forms. We obtain an integral identity and some Hermite-Hadamard-Fejér type integral inequalities for p-convex functions in fractional integral forms. We give some Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities for convex, harmonically convex and p-convex functions.

Main Results
Throughout this section, w k k 1 ¼ sup t2 a;b ½ wðtÞ j j, for the continuous function w : a; b ½ ! R.
holds for all x 2 a; b ½ .
This completes the proof of i. (ii) The proof is similar to i.
Theorem 9 Let f : ½ and w : a; b ½ ! R is nonnegative, integrable and psymmetric with respect to a p þb p 2 Â Ã 1=p , then the following inequalities for fractional integrals hold: : Multiplying both sides of (14) by and integrating with respect to t over 0; 1 ½ , using Lemma 1-i, we get the left hand side of (12). For the proof of the second inequality in (12), we first note that if f is a p-convex function, then, for all t 2 0; 1 ½ , it yields Multiplying both sides of (15) by and integrating with respect to t over 0; 1 ½ , using Lemma 1-i, we get the right hand side of (12). This completes the proof of i. (ii) The proof is similar to i.
h Remark 1 In Theorem 9, one can see the following.
Lemma 2 Let f : I & 0; 1 ð Þ!R be a differentiable function on I and a; b 2 I with a\b, p 2 Rn 0 f g and a [ 0. If f 0 2 L a; b ½ and w : a; b ½ ! R is integrable and psymmetric with respect to a p þb p 2 Â Ã 1=p , then the following equalities for fractional integrals hold: with gðxÞ ¼ x 1=p , x 2 a p ; b p ½ .

Proof
(i) Let p [ 0. It suffices to note that By integration by parts and using Lemma 1-i, we have and similarly A combination of (18), (19) and (20) gives (16). This completes the proof of i. (ii) The proof is similar to i.

Remark 2
In Lemma 2, one can see the following.
(1) If one takes p ¼ 1, one has İşcan (2015) Theorem 10 Let f : I & 0; 1 ð Þ!R be a differentiable function on I such that f 0 2 L a; b ½ , where a; b 2 I and a\b. If f 0 j j is p-convex function on a; b ½ for p 2 Rn 0 f g and a [ 0, w : a; b ½ ! R is continuous and p-symmetric with respect to a p þb p 2 Â Ã 1=p , then the following inequality for fractional integrals hold: where Iran J Sci Technol Trans Sci (2018) 42:2079-2089 2083 C 3 a; p ð Þ ¼ Since w is p-symmetric with respect to a p þb p 2 Â Ã 1=p , A combination of (21) and (22) gives Since f 0 j j is p-convex function on a; b ½ , we have A combination of (23) and (24) gives This completes the proof of i. (ii) The proof is similar to i.

Remark 3
In Theorem 10, one can see the following.
(1) If one takes p ¼ 1 and a ¼ 1, one has the following Hermite-Hadamard-Fejér inequality for convex functions: (2) If one takes p ¼ À1, one has the following Hermite-Hadamard-Fejér inequality for harmonically convex functions via fractional integrals: (3) If one takes p ¼ À1, a ¼ 1 and w x ð Þ ¼ 1, one has the following Hermite-Hadamard inequality for harmonically convex functions: (4) If one takes p ¼ À1 and a ¼ 1, one has the following Hermite-Hadamard-Fejér inequality for harmonically convex functions: (5) If one takes p ¼ À1 and w x ð Þ ¼ 1, one has the following Hermite-Hadamard inequality for harmonically convex functions via fractional integrals: Theorem 11 Let f : I & 0; 1 ð Þ!R be a differentiable function on I such that f 0 2 L a; b ½ , where a; b 2 I and a\b. If f 0 j j q , q ! 1, is p-convex function on a; b ½ for p 2 Rn 0 f g, a [ 0, w : a; b ½ ! R is continuous and psymmetric with respect to a p þb p 2 Â Ã 1=p , then the following inequality for fractional integrals holds: where C 1 a; p ð Þ, C 2 a; p ð Þ are the same in Theorem 10, where C 3 a; p ð Þ, C 4 a; p ð Þ are the same in Theorem 10, Using (23), power mean inequality and the p-convexity of f 0 j j q ; it follows that This completes the proof of i. (ii) The proof is similar to i.

Remark 4
In Theorem 11, one can see the following.
Corollary 2 In Theorem 11, one can see the following.
(1) If one takes p ¼ 1 and w x ð Þ ¼ 1, one has the following Hermite-Hadamard inequality for convex functions via fractional integrals: (2) If one takes p ¼ 1 and a ¼ 1, one has the following Hermite-Hadamard-Fejér inequality for convex functions: (3) If one takes p ¼ À1, one has the following Hermite-Hadamard-Fejér inequality for harmonically convex functions via fractional integrals: (4) If one takes p ¼ À1 and a ¼ 1, one has the following Hermite-Hadamard-Fejér inequality for harmonically convex functions: Theorem 12 Let f : I & 0; 1 ð Þ!R be a differentiable function on I such that f 0 2 L a; b ½ , where a; b 2 I and a\b. If f 0 j j q , q [ 1, is p-convex function on a; b ½ for p 2 Rn 0 f g, a [ 0, 1 q þ 1 r ¼ 1, w : a; b ½ ! R is continuous and p-symmetric with respect to a p þb p 2 Â Ã 1=p , then the following inequality for fractional integrals holds: f a