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.

We relate the connection between the sizes of circuits in suborbital graph for the normalizer of in PSL(2,) and the congruence equations arising from related group action. We give a number theoretic result which says that all prime divisors of for any integer u must be congruent to .

In this work, the concepts of quantum derivative and quantum integral are renamed to be the left quantum derivative and the left de.. . nite quantum integral. Symmetrically to the left, a new quantum derivative (the right) and de.. . nite quantum integral (the right) are de.. . ned. Some properties of these new concepts are investigated and as well as according todo these new concepts some inaccuracies in quantum integral inequalities are corrected. Moreover, some new quantum Hermite-Hadamard type inequalities are established. Hosted file M_Kunt_, A_ W_ Baidar_and_Z_\selectlanguage{polish}Ş\selectlanguage{english}anl\selectlanguage{polish}ı.

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 .

In this work, new quantum derivative and integral concepts are de…ned. Some properties of these new concepts are investigated. Using two symmetrically quantum integrals together, some new quantum Hermite-Hadamard type inequalities are established. Also, using a new quantum identity, some known trapezoid type inequalities are extended to quantum analogs.