Ostrowski Type Inequalities Involving Harmonically Convex Functions and Applications

The main objective of this paper is to derive some new generalizations of Ostrowski type inequalities for the functions whose first derivatives absolute value are harmonically convex. We also discuss some special cases of the obtained results. In the last section, we present some applications of the obtained results.

It is worth mentioning here that the harmonic property has played a significant role in different fields of pure and applied sciences. In [2], the authors have discussed the important role of the harmonic mean in Asian stock options. Interestingly, harmonic means are applied in electric circuit theory. More specifically, the total resistance of a set of parallel resistors is just half of the harmonic mean of the total resistors. For example, if R 1 and R 2 are the resistances of two parallel resistors, then the total resistance is computed by the formula: which is the half of the harmonic mean. Noor [3] showed that the harmonic mean also plays a crucial role in the development of parallel algorithms for solving nonlinear problems. Several authors have used harmonic means and harmonic convex functions to propose some iterative methods for solving linear and nonlinear system of equations.
Convexity played a vital role in the development of the theory of inequalities. Several inequalities constitute direct consequences of applications of convexity. In this regard, Hermite-Hadamard's inequality is one of the most extensively as well as intensively studied result. It provides us with an estimate of the (integral) mean value of a continuous convex function. It reads as: Let Λ : I = [a 1 , a 2 ] ⊂ R → R be a convex function, then Iscan [4] extended the classical version of Hermite-Hadamard's inequality using the harmonic convexity property of the function.
An interesting problem related to Hermite-Hadamard's inequality is its precision. Note that the left Hermite-Hadamard inequality can be estimated by the following inequality: where M is the Lipschitz constant which is equal to sup Λ(x)−Λ(y) x−y ; x = y . This above inequality is known in the literature as Ostrowski's inequality.
Recently, several research articles have been written on different generalizations of Ostrowski's inequality using different techniques. For example, Alomari et al. [5] obtained Ostrowski type inequalities using the class of s-convex functions. Ardic et al. [6] obtained Ostrowski type inequalities using the class of GG-convex and GA-convex functions. Budak and Sarikaya [7] obtained generalized Ostrowski type inequalities for functions whose first derivatives' absolute values are convex. Budak and Sarikaya [8] also obtained some new weighted Ostrowski type inequalities for functions of two variables with bounded variation. Iscan [4] obtained some Ostrowski type inequalities using the class of harmonically sconvex functions. Khurshid et al. [9] obtained a conformable fractional version of Ostrowski type inequalities using preinvex functions. Koroglu [10] obtained some more generalized Ostrowski type inequalities using harmonically convex functions. Mohsin et al. [11] obtained new generalizations of Ostrowski type inequalities using harmonically h-convex functions. Set [12] obtained some generalized fractional refinements of Ostrowski type inequalities using the class of s-convex functions. Recently, Sun [13] obtained some more local fractional versions of Ostrowski type inequalities and discussed its applications. For more details on Ostrowski type inequalities and its applications, cf. [14].
The aim of this paper is to obtain some new generalizations of Ostrowski's inequality essentially utilising the harmonic convexity property of the functions. We first derive a new auxiliary result which will play a significant role in the development of these results. We also discuss some special cases that can be deduced from the main results of the paper. In the last section, we present some applications of the obtained results. We hope that the ideas and techniques presented within this paper will inspire interested readers.

Main Results
In this section, we discuss our main results.
Let L[a 1 , a 2 ] be the space of Lebesgue integrable functions on the interval [a 1 , a 2 ]. Our first result is an auxiliary lemma. This result will play a significant role in the development of the next results.
If Λ belongs to L[a 1 , a 2 ], then, for all ∈ (0, 1) and for all x ∈ [a 1 , a 2 ], the following equality holds true: (1) Proof. For the sake of brevity, we write Now, an integration by parts yields Similarly, Summing I 1 and I 2 , we have This implies This completes the proof.
We now discuss some special cases of Lemma 1. To the best of our knowledge, these special cases are also new in the literature. I. If we set ς = 1 in Lemma 1, then the following equality holds: In addition, if we set = 0 in (3), then the following equality holds: where x(a 2 −a 1 ) , 1 .
Before proceeding to our next result, let us recall the integral form of the hypergeometric function: We now derive our first refinement of Ostrowski type inequality using the class of harmonically convex functions.

Proof. For the sake of brevity, we write
Using the harmonic convexity of |Λ |, we get This completes the proof.
We now derive a second refinement of Ostrowski type inequalities involving harmonically convex functions. We derive this result using the power mean inequality. Theorem 2. Let Λ : I → R be a differentiable mapping on I • , where a 1 , a 2 ∈ I with a 1 < a 2 . If, for q ≥ 1, |Λ | q is harmonically convex, for ∈ [0, 1], x ∈ [a 1 , a 2 ] and ς ∈ [0, 1] \ { 1 2 }, the following inequality holds true: where Proof. For the sake of brevity, we write A a 1 = ςa 1 + (1 − ς)a 2 and A a 2 = ςa 2 + (1 − ς)a 1 . We suppose that q > 1. Taking modulus in Lemma 1 and using the power mean inequality, we have Using the harmonic convexity of |Λ | q , we get This completes the proof.

Corollary 1.
Under the assumptions of Theorem 2, if we choose ς = 0 in (4), then we have where φ 1 , φ 2 are given in Theorem 2 and

Corollary 3.
Under the assumptions of Theorem 2, if we choose = 0 and x = 2a 1 a 2 a 1 +a 2 in Corollary 2, then In order to establish our next refinement of Ostrowski's inequality, we use Hölder's inequality.
Taking modulus in Lemma 1 and using Hölder's inequality, we have Using the harmonic convexity of |Λ | q , we get This completes the proof.