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

We consider a family of analytic and normalized functions with the property that zf′(z)/f(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$zf'(z)/f(z)$\end{document} (or 1+zf″(z)/f′(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1+zf''(z)/f'(z)$\end{document}) lies in a domain bounded by a right branch of a hyperbola ρ=ρ(s)=(2cosφs)−s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho =\rho (s)= ( 2 \cos \frac{\varphi }{s} )^{-s}$\end{document}, where 0<s≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< s\le 1$\end{document} and |φ|<(πs)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|\varphi |<(\pi s)/2$\end{document}. A comprehensive characteristic of that families and relations with the well known families of univalent functions are presented. Some relevant examples are indicated.