In this paper, firstly, new Hermite–Hadamard type inequalities for harmonically convex functions in fractional integral forms are given. Secondly, Hermite–Hadamard–Fejer inequalities for harmonically convex functions in fractional integral forms are built. Finally, an integral identity and some Hermite–Hadamard–Fejer type integral inequalities for harmonically convex functions in fractional integral forms are obtained. Some results presented here provide extensions of others given in earlier works.

Conformable fractional derivative is introduced by the authors Khalil et al. In this study we develop their concept and introduce multi-variable conformable derivative for a vector valued function with several variables

In this study, we investigate suborbital graphs G_u,n of the normalizer Γ_B (N) of Γ₀ (N) in PSL(2, ℝ) for N = 2^α3^β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of Γ_B (N) on $\stackrel{\wedge }{\mathbb{Q}}$ using the group ${\text{Г}}_{{\rm B}}^{+}$ (N) and then obtain some properties of the graphs arising from this action.

The present paper is devoted to studying the maps corresponding to the suborbital graphs for the normalizer ${\mathsf{\Gamma }}_B\left(N\right)$ of ${\mathsf{\Gamma }}_0\left(N\right)$ modulo N, where N denotes a positive integer. We reveal the complete structure of these maps, finding their vertices, edges, darts, and faces explicitly. The maps we investigated in the present paper were all regular maps of large genus except for some low values of N.