Ostrowski type Inequalities via h-convex Functions with Applications for Special Means

In this paper, we establish some new Ostrowski type inequalities for the class of h-convex functions which are super-multiplicative or super-additive and nonnegative. Some applications for special means and PDF's are given.


Introduction
[10] Let f : I ⊂ [0, ∞) → R be a differentiable mapping on I • , the interior of the interval I, such that f ′ ∈ L [a, b], where a, b ∈ I with a < b. If |f ′ (x)| ≤ M , then the following inequality: holds. This result is known in the literature as the Ostrowski inequality. For recent results and generalizations concerning Ostrowski's inequality, see [1,2,3,8,5,11] and the references therein. Definition 1. [12] We say that f : I → R is Godunova-Levin function or that f belongs to the class Q (I) if f is non-negative and for all x, y ∈ I and t ∈ (0, 1) we have Definition 2. [9] We say that f : I ⊆ R → R is a P -function or that f belongs to the class P (I) if f is nonnegative and for all x, y ∈ I and t ∈ [0, 1] , we have for all x, y ∈ [0, ∞) and t ∈ [0, 1]. This class of s-convex functions is usually denoted by K 2 s . Definition 4. [16] Let h : J → R be a nonnegative function, h ≡ 0. We say that f : I ⊆ R → R is h-convex function, or that f belongs to the class SX (h, I), if f is nonnegative and for all x, y ∈ I and t ∈ [0, 1] we have If inequality (1.5) is reversed, then f is said to be h-concave, i.e. f ∈ SV (h, I). Obviously, if h (t) = t, then all nonnegative convex functions belong to SX (h, I) and all nonnegative concave functions belong to SV (h, I); if h (t) = 1 t , then SX (h, I) = Q (I); if h (t) = 1, then SX (h, I) ⊇ P (I); and if h (t) = t s , where s ∈ (0, 1), then SX (h, I) ⊇ K 2 s . Remark 1. [16] Let h be a non-negative function such that for all α ∈ (0, 1). For example, the function h k (x) = x k where k ≤ 1 and x > 0 has that property. If f is a non-negative convex function on I , then for x, y ∈ I , α ∈ (0, 1) we have .
Similarly, if the function h has the property: h(α) ≤ α for all α ∈ (0, 1), then any non-negative concave function f belongs to the class SV (h, I).

Definition 5. [16]
A function h : J → R is said to be a super-multiplicative function if for all x, y ∈ J, when xy ∈ J.
If inequality (1.8) is reversed, then h is said to be a sub-multiplicative function. If equality is held in (1.8), then h is said to be a multiplicative function.
for all x, y ∈ J, when x + y ∈ J.
In [15], M.Z. Sarıkaya, A. Saglam and H. Yıldırım established the following Hadamard type inequality for h-convex functions: For recent results related h-convex functions see [6,7,14,15,16]. The aim of this study is to establish some Ostrowski type inequalities for the class of functions whose derivatives in absolute value are h-convex and h-concave functions.

Ostrowski type inequalities for h-convex functions
In order to achieve our objective, we need the following lemma [8]: , then the following equality holds; Theorem 2. Let h : J ⊆ R → R be a nonnegative and super-multiplicative functions, f : Proof. By Lemma 1 and since |f ′ | is h-convex, then we can write; The proof is completed.
In the next corollary, we will also make use of the Beta function of Euler type, which is for x, y > 0 defined as One of the important result is given in the following theorem.
Theorem 3. Let h : J ⊆ R → R be a nonnegative and superadditive functions, Proof. Suppose that p > 1. From Lemma 1 and using the Hölder's inequality, we can write Since |f ′ | q is h-convex and by using properties of h-convexity in the assumptions, Similarly, we can show that Therefore, we obtain The proof is completed.
For example, h (t) = t 2 is a superadditive function for nonnegative real numbers because the square of (u + v) is always greater than or equal to the square of u plus the square of v, for u, v ∈ [0, ∞).
Remark 3. Since 1 np+1 1 p < 1 2 , for any 4 ≥ n > p > 1, n ∈ N, then we behold that the inequality (2.3) is better than the inequality (1.1). Better approaches can be obtained even it is irregular for bigger n and p numbers.
As we know, h-convex functions include all nonnegative convex, s-convex in the second sense, Q(I)-convex and P -convex function classes. In this respect, it is normal to obtain weaker results once compared with inequalities in referenced studies. Because, the inequalities written herein were considered to be more general than above-mentioned classes and it was taken into account to be super-multiplicative or super-additive material. In this case, right side of inequality may be greater.
A new approach for h-convex function is given in the following result.
Theorem 4. Let h : J ⊆ R → R be a nonnegative and supermultiplicative func- Proof. Suppose that q ≥ 1. From Lemma 1 and using the power mean inequality , we have Similarly, we can observe that Therefore, we deduce and the proof is completed.
Remark 4. i) In the above inequalities, one can establish several midpoint type inequalities by letting x = a+b 2 .
ii) In Theorem 4, if we choose (a) x = a+b 2 , then we obtain The following result holds for h-concave functions.
Theorem 5. Let h : J ⊆ R → R be a non-negative and superadditive functions, for each x ∈ [a, b] .
Proof. Suppose that p > 1. From Lemma 1 and using the Hölder's inequality, we can write But since |f ′ | q is h-concave, using the inequality (1.10), we have