Abstract. The author introduce the concept of harmonically convex functions and establish some Hermite-Hadamard type inequalities of these classes of functions. IntroductionLet f : I ⊂ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The following inequalityholds. This double inequality is known in the literature as Hermite-Hadamard integral inequality for convex functions. Note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the mapping f . Both inequalities hold in the reversed direction if f is concave. For some results which generalize, improve and extend the inequalities(1.1) we refer the reader to the recent papers (see [1,2,4,3,5] ).The main purpose of this paper is to introduce the concept of harmonically convex functions and establish some results connected with the right-hand side of new inequalities similar to the inequality (1.1) for these classes of functions. Some applications to special means of positive real numbers are also given. Main ResultsDefinition 1. Let I ⊂ R\ {0} be an real interval. A function f : I → R is said to be harmonically convex, iffor all x, y ∈ I and t ∈ [0, 1]. If the inequality in (1.1) is reversed, then f is said to be harmonically concave.Example 1. Let f : (0, ∞) → R, f (x) = x, and g : (−∞, 0) → R, g(x) = x, then f is a harmonically convex function and g is a harmonically concave function.The following proposition is obvious from this example:Proposition 1. Let I ⊂ R\ {0} be an real interval and f : I → R is a function, then ;2000 Mathematics Subject Classification. Primary 26D15; Secondary 26A51.

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The celebrated Hermite–Hadamard and Ostrowski type inequalities have been studied extensively since they have been established. We find novel versions of the Hermite–Hadamard and Ostrowski type inequalities for the n-polynomial s-type convex functions in the frame of fractional calculus. Taking into account the new concept, we derive some generalizations that capture novel results under investigation. We present two different general techniques, for the functions whose first and second derivatives in absolute value at certain powers are n-polynomial s-type convex functions by employing $\mathcal{K}$ K -fractional integral operators have yielded intriguing results. Applications and motivations of presented results are briefly discussed that generate novel variants related to quadrature rules that will be helpful for in-depth investigation in fractal theory, optimization and machine learning.