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 by İşcan (Hacet J Math Stat 46(6):935–942, 2014).

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 differentiable functions. By using these identities, we obtain some new trapezoidal type inequalities for harmonically convex functions. Our results generalize many results from earlier papers.

We relate the connection between the sizes of circuits in suborbital graph for the normalizer of \(\Gamma _0(m)\) in PSL(2,\(\mathbb {R}\)) and the congruence equations arising from related group action. We give a number theoretic result which says that all prime divisors of \(3u^2\mp 3u+1\) for any integer u must be congruent to \(1\pmod {3}\).

In this paper we show that the subgraph F 3 is disconnected and that for all integers m, we find all integers a and b such that ( 9 m 2 − 4 ) a 2 + 4 and 5 b 2 ± 4 are square. It turns out that the set of numbers b comprises the Fibonacci numbers.

This paper establishes a weighted Simpson-type identity on the parameter domain associated with quantum integral operators. Using this identity together with Hölder’s inequality and the power mean inequality, we derive new estimates for classes of functions whose associated parameter-domain q-derivatives satisfy h-convexity assumptions. Additional bounds are obtained under boundedness and Lipschitz conditions. Applications to the s-moment of a random variable and to several special means are derived in the classical limit $q\to 1^{-}$.

We establish a connection between suborbital graphs and integer solutions of Pell?s equation of the form x2-Ny2 = 1, where N is a non-square positive integer. We derive new suborbital graphs generated by the action of some specific modular subgroups on extended rational numbers. By using these graphs, we obtain a new combinatorial notation for the integer solutions of Pell?s equation and some results on the vertices of the graphs studied here.