We establish some inequalities of Schwarz–Pick type for harmonic and hyperbolic functions on the unit ball $${\mathbb {R}^n}$$ we disprove a recent conjecture Liu (Int Math Res Not, 2021).

Two classes of univalent harmonic functions on unit disc satisfying the conditions ∑∞ n=2(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) and ∑∞ n=2 n(n−α)(|an|+|bn|) ≤ (1−α)(1−|b1|) are given. That the ranges of the functions belonging to these two classes are starlike and convex, respectively. Sharp coefficient relations and distortion theorems are given for these functions. Furthermore results concerning th...

In this paper , certain subclass of harmonic multivalent function defined in the exterior unit disk by used generalize hypergeometric functions is introduced . This study an attempting have been made to investigate several geometric properties such as coefficient property growth bounds extreme points convolution and convex linear combination