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Inequalities via s-convexity and log-convexity

In this paper, we obtain some new inequalities for functions whose second derivatives' absolute value is s-convex and log-convex. Also, we give some applications for numerical integration.


INTRODUCTION
We start with the well-known definition of convex functions: a function f : I → R, ∅ = I ⊂ R, is said to be convex on I if inequality holds for all x, y ∈ I and t ∈ [0, 1] .
In the paper [12], authors gave the class of functions which are s−convex in the second sense by the following way. A function f : [0, ∞) → R is said to be s−convex in the second sence if holds for all x, y ∈ [0, ∞), t ∈ [0, 1] and for some fixed s ∈ (0, 1]. The class of s−convex functions in the second sense is usually denoted with K 2 s . Besides in [12], Hudzik and Maligranda proved that if s ∈ (0, 1) f ∈ K 2 s implies f ([0, ∞)) ⊆ [0, ∞), i.e., they proved that all functions from K 2 s , s ∈ (0, 1) , are nonnegative.
It can be easily checked that (i) If b ≥ 0 and 0 ≤ c ≤ a, then f ∈ K 2 s , (ii) If b > 0 and c < 0, then f / ∈ K 2 s . Several researchers studied on s−convex functions, some of them can be found in [12]- [17].
Another kind of convexity is log −convexity that is mentioned in [6] by Niculescu as following.
A positive function f is called log −convex on a real interval I = [a, b], if for all x, y ∈ [a, b] and λ ∈ [0, 1], For recent results for log −convex functions, we refer to readers [2]- [9]. Now, we give a motivated inequality for convex functions: Let f : I ⊂ R → R be a convex function on the interval I of real numbers and a, b ∈ I with a < b. The inequality is known as Bullen's inequality for convex functions [8], p. 39. We also consider the following useful inequality: then the following inequality holds (see [11]).
This inequality is well known in the literature as the Ostrowski inequality.
The main aim of this paper is to prove some new integral inequalities for s−convex and log −convex functions by using the integral identity that is obtained by Sarıkaya and Set in [1]. We also give some applications to our results in numerical integration. Some of our results are similar to the Ostrowski inequality and for special selections of the parameters, we proved some new inequalities of Bullen's type.

inequalities for s−convex functions
We need the following Lemma which is obtained by Sarıkaya and Set in [1], so as to prove our results: where α, β ∈ R nonnegative and not both zero, then the identity holds where α, β ∈ R nonnegative and not both zero.
Proof. From Lemma 1, using the property of the modulus and s− convexity of |f ′′ | , we can write where we use the fact that The proof is completed.

Corollary 1. Suppose that all the assumptions of Theorem 1 are satisfied with
Corollary 3. In Theorem 1, if we choose α = β = 1 2 and x = a+b 2 , we obtain the following Bullen type inequality; Theorem 2. Let f : [a, b] → R be an absolutely continuous mapping such that x, y > 0 is the Euler Beta function, α, β ∈ R nonnegative and not both zero.
Proof. From Lemma 1, using the property of the modulus, Hölder inequality and s−convexity of |f ′′ | q , we can write We get the desired result by making use of the necessary computation.
Theorem 3. Under the assumptions of Theorem 2, the following inequality holds where β (x, y) is the Euler Beta function.
Proof. From Lemma 1, using the property of the modulus, Hölder inequality and s−convexity of |f ′′ | q , we can write We get the desired result by making use of the necessary computation.
The next result is obtained by using the well-known power-mean integral ineqaulity: holds where α, β ∈ R nonnegative and not both zero.
Proof. From Lemma 1, using the property of the modulus, power-mean integral inequality and s−convexity of |f ′′ | q , we can write The proof is completed.

inequalities for log −convex functions
In this section, we will give some results for log −convex functions. For the simplicity, we will use the following notations: Theorem 5. Let f : [a, b] → R be an absolutely continuous mapping such that holds where κ = 1, τ = 1, α, β ∈ R nonnegative and not both zero.
Proof. From Lemma 1 and by using the log − convexity of |f ′′ | , we can write By a simple computation, we get the result.
Proof. From Lemma 1, by using log −convexity of |f ′′ | q and by applying Hölder inequality, we get By computing the above integrals, we get the desired result.
Proof. From Lemma 1, by using the well-known power-mean integral inequality and log −convexity of |f ′′ | q , we have Which completes the proof.