Structural formulas and value regions of functionals in certain classes of regular functions

E. G. Goluzina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 11, Tome 204 (1993), pp. 55-60

Résumé

We study the structural properties of the class $M_{k,\lambda,b}$ ($k\ge2$, $0\le\lambda\le1$, $b\in\mathbb C\setminus\{0\}$) of functions $f(z)=z+\dots$ which are regular in $|z|<1$ and satisfy the conditions $f(z)f'(z)z^{-1}\ne0$ and $\lim_{r\to1-0}\int_0^{2\pi}|\operatorname{Re}J(z)|\,d\theta\le k\pi$ ($z=re^{i\theta}$), where $$ J(z)=\lambda(1+b^{-1}zf''(z)/f'(z))+(1-\lambda)(b^{-1}zf'(z)/f(z)+1+b^{-1}). $$ The value regions of some functionals on this class are found. The case $\lambda=1$ was considered in our previous paper. Bibliography: 4 titles.