On Some New Hermite-Hadamard Type Inequalities for s-Geometrically Convex Functions

Let f : I ⊂ R + = (0,∞) → [1,∞). It is easy to show that f is s-geometrically convex function on [a, b], a, b ∈ I with a < b if and only if ln(f ∘ exp) is s-convex function on [ln a, ln b]. Let f : I ⊆ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The following double inequality is well known in the literature as Hermite-Hadamard integral inequality:


Introduction
In this section, we firstly list several definitions and some known results. Let f : I ⊆ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The following double inequality is well known in the literature as Hermite-Hadamard integral inequality the inequalities (1.1) for the s-geometrically convex and monotonically decreasing functions. In [6], Tunc has established inequalities for s-geometrically and geometrically convex functions which are connected with the famous Hermite Hadamard inequality holding for convex functions. In [6], Tunc also has given the following result for geometrically convex and monotonically decreasing functions: Corollary 1. Let f : I ⊂ R + → R + be geometrically convex and monotonically decreasing on [a, b], then one has Note that, the inequalities (1.2) are also true without the condition monotonically decreasing and the inequalities (1.2) are sharp.
In this paper, the author give new identities for differentiable functions. A consequence of the identities is that the author establish some new inequalities connected with the inequalities (1.2) for the s-geometrically convex functions.

Main Results
In order to prove our results, we need the following lemma: Proof. Integrating by part and changing variables of integration yields By the following equality, we obtain the inequality (2.1) This completes the proof of Lemma 1.
Theorem 1. Let f : I ⊆ R + → R + be differentiable on I • , and a, b ∈ I • with a < b and f ′ ∈ L [a, b] . If |f ′ | q is s-geometrically convex on [a, b] for q ≥ 1 and s ∈ (0, 1] , , from lemma 1 and power mean inequality, we have . Since |f ′ | q is s-geometrically convex on [a, b], from lemma 1 and Hölder inequality, we have From (2.14) to (2.18), (2.4) holds. This completes the required proof.
If taking s = 1 in Theorem 1, we can derive the following corollary.
where θ, ϑ, H 1 , H 2 , h 1 and h 2 are the same as in Theorem 1.
If taking q = 1 in Theorem 1, we can derive the following corollary.
Corollary 3. Let f : I ⊆ R + → R + be differentiable on I • , and a, b ∈ I • with a < b where θ, ϑ, H 1 , H 2 , h 1 and h 2 are the same as in Theorem 1. (2.20) , and θ, ϑ are the same as in (2.6).
. Since |f ′ | q is s-geometrically convex on [a, b], from lemma 1 and Hölder inequality, we have where θ, ϑ, H 3 and h 3 are the same as in Theorem 2.