Relations of a planar domains bounded by hyperbolas with families of holomorphic functions

Abstract

Abstract We consider a family of analytic and normalized functions with the property that zf′(z)/f(z) $zf'(z)/f(z)$ (or 1+zf″(z)/f′(z) $1+zf''(z)/f'(z)$) lies in a domain bounded by a right branch of a hyperbola ρ=ρ(s)=(2cosφs)−s $\rho =\rho (s)= ( 2 \cos \frac{\varphi }{s} )^{-s}$, where 0<s≤1 $0< s\le 1$ and |φ|<(πs)/2 $|\varphi |<(\pi s)/2$. A comprehensive characteristic of that families and relations with the well known families of univalent functions are presented. Some relevant examples are indicated.

Keywords

• Univalent functions
• Subordination
• Starlike and convex functions
• Domain bounded by hyperbola