Some generalized inequalities of Hermite-Hadamard type for strongly s-convex functions

Both inequalities hold in the reversed direction if f is concave. We note that Hadamards inequality may be regarde d s a refinement of the concept of convexity and it follows easily f rom Jensens inequality. Hadamards inequality for convex functions has received renewed attention in recent years an d a remarkable variety of refinements and generalizations have been found (see, for example, [ 2] [5],[11],[16],[18]) and the references cited therein.


Introduction
In this section, we firstly list several definitions and some known results. Many inequalities have been established for convex functions but the most famous inequality is the Hermite-Hadamards inequality, due to its rich geometrical significance and applications( [4], [12, p.137]). These inequalities state that if f : I ⊆ R → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b, then Both inequalities hold in the reversed direction if f is concave. We note that Hadamards inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensens inequality. Hadamards inequality for convex functions has received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found (see, for example, [2], [5], [11], [16], [18]) and the references cited therein.
In [6], Hudzik and Maligranda considered, among others, the class of functions which are s-convex in the second sense.
To prove our main results, we consider the following Lemmas given by Sarikaya et al. in [14] and Kiriş and Sarikaya in [7], respectively: for n ∈ a, a+b 2 and m ∈ a+b 2 , b .
The aim of the paper is to establish some new generalized Hermite-Hadamard inequalities for function whose derivatives absolute values are strongly s−convex.

Main results
Firstly, we will give some calculated integrals which used our main results: Theorem 2. Let f : for some s ∈ (0, 1] with modulus c > 0 , then following inequality holds : 3 12 for n ∈ a, a+b 2 and m ∈ a+b 2 , b .
Proof. Taking modulus in Lemma 1 and using the strongly s−convexity of | f ′ |, we have If we substitute the equalities (6)- (11) in (13), then we obtain required result (12).
Remark. If we choose m = b, n = a in Theorem 2, then we have Remark. If we choose m = n = a+b 2 in Theorem 2, then we have Remark. If we choose m = a+5b 6 , n = 5a+b 6 in Theorem 2, then we have For c = 0 it reduces to the Hermite-Hadamard-type inequalities for s-convex functions proved by Sarikaya et al. in [21].
Proof. From Lemma 1 and by using the Hölder inequality , then we have Using the strongly s-convexity of | f ′ | q , we have By simple computation, we have 3 12 .

Corollary 2.
If we choose m = b and n = a in Theorem 3, then we have 2  Remark. Choosing s = 1 in Corollary 2, we obtain the inequality Corollary 3. If we choose m = n = a+b 2 in Theorem 3, then we have