Some Ostrowski type integral inequalities via generalized harmonic convex functions

C urrently the theory of convexity is a hot filed of research due to its applications in optimization theory. This theory also provides the solution to many problems which arise in applied and pure mathematics. This theory plays a remarkable role in the study of linear and non–linear programming problems. In past years, this theory attracts many mathematicians, see [1–3]. In mathematics, the idea of convexity is based on the theory of inequalities. The current perspective on the term inequalities require a robust and interconnection between geometry and analysis, which makes the reader shear a sense of exhilaration and exultation. For the applications interested readers can see [4,5]. In 1938, A. Ostrowski established an inequality namely Ostrowski inequality. Mathematically Ostrowski inequality [6] is given by, let Ψ : A ⊆ R→ R is an differential function on A◦ and ℘1,℘2 ∈ A◦ with ℘1 < ℘2. If |Ψ(x)| 6 M, ∀x ∈ [℘1,℘2], then we attain the following inequality ∣∣∣∣Ψ(x)− 1 ℘2 − ℘1 ∫ ℘2


Introduction
C urrently the theory of convexity is a hot filed of research due to its applications in optimization theory.
This theory also provides the solution to many problems which arise in applied and pure mathematics. This theory plays a remarkable role in the study of linear and non-linear programming problems. In past years, this theory attracts many mathematicians, see [1][2][3].
In mathematics, the idea of convexity is based on the theory of inequalities. The current perspective on the term inequalities require a robust and interconnection between geometry and analysis, which makes the reader shear a sense of exhilaration and exultation. For the applications interested readers can see [4,5].
In 1938, A. Ostrowski established an inequality namely Ostrowski inequality. Mathematically Ostrowski inequality [6] is given by, let Ψ : A ⊆ R → R is an differential function on A • and ℘ 1 , ℘ 2 ∈ A • with ℘ 1 < ℘ 2 . If |Ψ (x)| M, ∀x ∈ [℘ 1 , ℘ 2 ], then we attain the following inequality This inequality has a great contribution and various applications in probability, numerical integration, and mathematical analysis. The 1 st time extension and multivariate of this inequality was investigated by Milovanović and Peçarić in [7] and Milovanović in [8] respectively. The Dragomir and Wang [9] explored an inequality of Ostrowski-Grüss type. From the last few decades, due to the effectiveness and applications, many authors generalized the Ostrowski's inequality, see [10][11][12].
In this paper, the refinements of Ostrowski type inequalities for s-type convex functions are investigated, which are the generalized and extended variants of the previously known results.
Recently Rashid et al., [16] introduced s-type convex function as follows: The term "hypergeometric function" sometimes refers to the generalized hypergeometric function. In 1949, Arnold Sommerfeld [17] introduced the hypergeometric function by the differential equation. In 1960, Kratzer and Franz determined the hypergeometric function in the support of hypergeometric series. In 1963, Lebedev investigated the hypergeometric function through the Γ-function and its logarithmic derivative. In 2006, Françoise et al., [18] introduced the Gaüss hypergeometric function defined for |z| < 1 by the hypergeometric series. This function is often represented by F (a, b; c; z). Mathematically hypergeometric function is given as [19]: This functions have a wide range of applications in partitions, combinatorial identities, cylinder and spherical problems, finite vector spaces, number theory, and physics.
Motivated by ongoing research activities in this direction, first of all in Section 3, we will explore the concept of harmonic s-type convex function. In addition, some examples in the manner of this newly introduced definition are elaborated. Furthermore, in Section 4, we investigate some refinements of the Ostrowski type inequality. Finally, we will give a conclusion.

Generalized harmonic convex function and its properties
The principal objective of this section is to define the new class of convex functions which are called harmonic s-type convex functions.

Remark 1.
Taking s = 1 in Definition 3, we attain the definition of harmonically convex function which is defined byİşcan [15].
for all κ ∈ [0, 1] and s ∈ [0, 1] and applying the property of harmonic convex function, we have Proof. Applying the property of harmonic s-type convex function and using the condition Now we give some examples.

Example 1.
As Ψ(ς) = e ς is an increasing convex function so it is harmonically convex function (see [20]). By using Proposition 1, it is a harmonic s-type convex function.

Example 2.
As Ψ(ς) = ln ς is an increasing convex function so it is harmonically convex function (see [20]). By using Proposition 1, it is a harmonic s-type convex function.

Refinements of Ostrowski type inequality via harmonic s-type convex functions
In this section, we are going to attain some refinements of Ostrowski type inequality by the way of harmonic s-type convex function. Throughout this section, diff mapp means differentiable mapping. In order to obtain the results, we need the following lemma: If the mapping |Ψ | l is harmonic s-type convex for l 1 and s ∈ [0, 1] , then Proof. Applying Lemma 2, properties of modulus, power mean inequality and the property of harmonic s-type convex function of |Ψ | l , we have The proof is complete.

Corollary 1.
In Theorem 2, taking |Ψ (x)| M, ∀ x ∈ [℘ 1 , ℘ 2 ], we attain the following inequality If the mapping |Ψ | l is harmonic s-type convex for l 1 and s ∈ [0, 1] then for all x ∈ [℘ 1 , ℘ 2 ], one has Proof. Applying Lemma 2, properties of modulus, power mean inequality and the property of harmonic s-type convex function of |Ψ | l , we have The proof is complete.

Corollary 2.
In Theorem 2, taking |Ψ (x)| M, ∀ x ∈ [℘ 1 , ℘ 2 ], we attain the following inequality If the mapping |Ψ | l is harmonic s-type convex for l 1 and s ∈ [0, 1] then for all x ∈ [℘ 1 , ℘ 2 ], one has Proof. Applying Lemma 2, properties of modulus, power mean inequality and the property of harmonic s-type convex function of |Ψ | l , we have The proof is complete.