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 $\frac{1}{\rho }={\left(2\cos \frac{\phi }{s}\right)}^s(0<s\le 1,|\phi |<(\pi s)/2$). We mainly study a coefficient problem of the families of functions for which $z{f}^{\prime }/f$ or $1+z{f}^{\prime \prime }/f^{\prime }$ map the unit disk onto a subset of $\mathbb{H}(s)$. We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.