  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.
