A geometric property of functions starlike of order $\alpha$

We determined the maximum radius $\delta_\alpha(r)$ of the disk $\Omega_r$, possessing the property that every function $f(z)$, starlike of order $\alpha$, is starlike in $|z|<r$ with respect to any point of $\Omega_r$. The problem is reduced to that of finding the minimum of a certain functional for which extremal function is determined.