In this paper, we proved two new Riemann-Liouville fractional Hermite-Hadamard type inequalities for harmonically convex functions using the left and right fractional integrals independently. Also, we have two new Riemann-Liouville fractional trapezoidal type identities for differentiable functions. Using these identities, we obtained some new trapezoidal type inequalities for harmonically convex functions. Our results generalize the results given byİşcan (Hacet J Math Stat 46(6):935-942, 2014). Mathematics Subject Classification 26A51 • 26A33 • 26D10 for all x, y ∈ I and t ∈ [0, 1]. If the inequality in (1.2) is reversed, then f is said to be harmonically concave.

The Fibonacci sequence has many interesting properties and studied by many mathematicians. The terms of this sequence appear in nature and is connected with combinatorics and other branches of mathematics. In this paper, we investigate the orbit of a special subgroup of the modular group. Taking0, we determined the orbit {T r c (∞) : r ∈ N}. Each rational number of this set is the form Pr(c)/Qr(c), where Pr(c) and Qr(c) are the polynomials in Z[c]. It is shown that Pr(1), and Qr(1) the sum of the coefficients of the polynomials Pr(c) and Qr(c) respectively, are the Fibonacci numbers, where Pr(c) = r s=0 2r − s s c 2r−2s + r s=1 2r − s s − 1 c 2r−2s+1 and Qr(c) = r s=1 2r − s s − 1 c 2r−2s+2 .

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In this paper, we prove three new Katugampola fractional Hermite-Hadamard type inequalities for harmonically convex functions by using the left and the right fractional integrals independently. One of our Katugampola fractional Hermite-Hadamard type inequalities is better than given in [17]. Also, we give two new Katugampola fractional identities for di¤erentiable functions. By using these identities, we obtain some new trapezoidal type inequalities for harmonically convex functions. Our results generalize many results from earlier papers.

Abstract:In this work, we pose a new equality for differentiable functions. By using this equality, we have some new Ostrowki type inequalities and some error estimates for the midpoint formula for functions whose derivatives in absolute values at certain powers are harmonically convex.