Estimates for conformal radius and distortion theorems for univalent functions

L. V. Kovalev

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 16, Tome 263 (2000), pp. 141-156

Résumé

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 $S_M=\{f\in S:|f(z)|<M\text{ for }|z|<1\}$.