FUZZY OSTROWSKI TYPE INEQUALITIES VIA h−CONVEX

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. We would like to state well-known Ostrowski inequality via h−convex by using the Fuzzy Reimann integrals. In addition, we establish some Fuzzy Ostrowski type inequalities for the class of functions whose derivatives in absolute values at certain powers are h−convex by Hölder’s and power mean inequalities. This class of h−convex function, which is the generalization of many important classes including class of Godunova-Levin s−convex, s−convex in the 2nd kind and hence contains convex functions. It also contains class of P−convex and class of Godunova-Levin. In this way we also capture the results with respect to convexity of functions.


INTRODUCTION
In recent years, the generalization of classical convex function have emerged resulting in applications in the field of Mathematics. From literature, we recall some definitions for different types of convex functions.
Remark 1.8. In Definition 1.7, one can see the following.
, then we get the concept of s−convex in 2 nd kind.
(5) If h(t) = t in (1.1), then we get the concept of ordinary convex.
Next we present the clasical ostrowski inequality.
then we say that φ is Fuzzy-Riemann integrable to ϕ ∈ F R , we write it as

FUZZY OSTROWSKI TYPE INEQUALITIES VIA h−CONVEX FUNCTIONS
In order to prove our main results, we need the following lemma that has been obtained in [5].
, then for x ∈ (a, b) the following identity holds: We make use of the beta function of Euler type, which is for x, y > 0 defined as Suppose all the assumptions of Lemma 2.1 hold. Additionally, λ ∈ (0, 1], φ : Then ∀x ∈ (a, b) the following inequality holds: Proof. From the Lemma 2.1, (2) If one takes h(t) = t s where s ∈ (0, 1] in (2.2), then one has the Fuzzy Ostrowski inequality for s−convex functions in 2 nd kind: (3) If one takes h(t) = 1 in (2.2), then one has the Fuzzy Ostrowski inequality for P−convex function: (4) If one takes h(t) = t in (2.2), then one has the Fuzzy Ostrowski inequality for convex function: in in (2.2), then one has the Fuzzy Ostrowski inequality for MT −convex function:  ∈ (a, b) the following inequality holds:
Corollary 2.7. In Theorem 2.6, one can see the following.
(1) If one takes h(t) = t −s where s ∈ [0, 1) in (2.10), then one has the Fuzzy Ostrowski inequality for Godunova-Levin s−convex functions: (2) If one takes h(t) = t s , where s ∈ (0, 1] in (2.10), then one has the Fuzzy Ostrowski inequality for s−convex functions in 2 nd kind: (3) If one takes h(t) = 1, in (2.10), then one has the Fuzzy Ostrowski inequality for P−convex function: (4) If one takes h(t) = t, in (2.10), then one has the Fuzzy Ostrowski inequality for convex function:

Fuzzy Ostrowski type midpoint inequalties via h−convex functions.
Remark 2.8. In Theorem 2.4, one can see the following.
(1) If one takes x = a+b 2 in (2.6), then one has the Fuzzy Ostrowski Midpoint inequality for h−convex function: (2) If one takes x = a+b 2 and h(t) = t −s where s ∈ [0, 1) in (2.6), then one has Fuzzy Ostrowski Midpoint inequality for Godunova-Levin s−convex functions: (3) If one takes x = a+b 2 and h(t) = t s where s ∈ [0, 1] in (2.6), then one has Fuzzy Ostrowski Midpoint inequality for s−convex functions in 2 nd kind: (4) If one takes x = a+b 2 and h(t) = 1 in (2.6), then one has the Fuzzy Ostrowski Midpoint inequality for P−convex function: (5) If one takes x = a+b 2 and h(t) = t in (2.6), then one has the Fuzzy Ostrowski Midpoint inequality for convex function: in (2.6), then one has the Fuzzy Ostrowski inequality for MT −convex function: Remark 2.9. In Theorem 2.6, one can see the following.
(1) If one takes x = a+b 2 in (2.10), one has the Fuzzy Ostrowski Midpoint inequality for h−convex function: (2) If one takes x = a+b 2 and h(t) = t −s where s ∈ [0, 1) in (2.10), then one has the Fuzzy Ostrowski Midpoint inequality for Godunova-Levin s−convex functions: