Coefficients problems for families of holomorphic functions related to hyperbola

Abstract

We consider a family of analytic and normalized functions that are related to the domains $\mathbb{H}(s)$, with a right branch of a hyperbolas $H(s)$ as a boundary. The hyperbola $H(s)$ is given by the relation $\frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0<s\le 1,\ |\varphi|<(\pi s)/2$). We mainly study a coefficient problem of the families of functions for which $zf'/f$ or $1+zf''/f'$ map the unit disk onto a subset of $\mathbb{H}(s)$. We find coefficients bounds, solve Fekete-Szeg\"{o} problem and estimate the Hankel determinant.