Coefficients problems for families of holomorphic functions related to hyperbola

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 $\frac{1}ρ=\left( 2\cos\fracφ{s}\right)^s\quad (0<s\le 1,\ |φ|<(πs)/2$). 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-Szegö problem and estimate the Hankel determinant.

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