On extremal problems in classes of univalent functions which do not assume given values

E. G. Emel'yanov ; G. V. Kuz'mina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 6, Tome 144 (1985), pp. 94-114

Résumé

Section 1 of the paper is devoted to extremal problems in the classes of conformal homeomorphisms of the circle and the annulus, connected directly with the problem on the maximum of the conformal modulus in the family of doubly connected domains. In Secs. 2 and 3 one considers the class $R$ of functions $f(\zeta)=c_1\zeta+c_2\zeta^2+\dotsb$ regular and univalent in the circle $U=\{|\zeta|1\}$ and such that $f(\zeta_1)f(\zeta_2)=1$ for $\zeta_1,\zeta_2\in U$ (the class of Bieberbach–Eilenberg functions). Here one solves the problem of the maximum of $|f^\prime(\zeta_0)|$ in the class of functions $f(\zeta)\in R$ with a fixed value $f(\zeta_0)$, where $\zeta_0$ is an arbitrary point $U$, and of the maximum of $|f^\prime(\zeta_0)|$ in the entire class $R$. For the proof one makes use of the method of the moduli of families of curves.