Some estimates in the class of analytic functions of bounded type

V. P. Vazhdaev ; S. A. Gel'fer

Sbornik. Mathematics, Tome 13 (1971) no. 2, pp. 267-284

Résumé

We consider the class $A_M$ of functions regular in the disk $|\zeta|1$ which for any $r$, $0\leqslant r1$, satisfy the condition $$ \int_0^{2\pi}\ln^+|f(re^{i\theta})|\,d\theta\leqslant2\pi M, $$ where $M$ does not depend on the function. Using a parametric representation of this class, the authors find exact estimates of the mean arithmetic value and the mean geometric value of the modulus of the function at equally spaced points of the circumference, estimates of the moduli and arguments of the function, the moduli of the derivatives and other values for the class $A_M$ and certain of its subclasses. The solution of these problems is based on variation formulas introduced earlier by one of the authors (RZhMat., 1967, 11B99). Bibliography: 14 titles.