Estimates for conformal radius and distortion theorems for univalent functions

A simple proof of the recent result by E. G. Emel'yanov concerning the maximum of the conformal radius $r(D,1)$ for a family of simply connected domains with a fixed value $r(D,0)$ is given. A similar problem is solved for a family of convex domains. Exact estimates for functionals of the form $|g'(w)|/|g(w)|^{\delta}$ are obtained for families of functions inverse to elements of the classes $S$ and $S_m$, where $S=\{f:f\text{ is regular and univalent in the disk }\{z:|z|<1\}\text{ and }f(0)=f'(0)-1=0\}$ and $S_M=\{f\in S:|f(z)|<M\text{ for }|z|<1\}$.