# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## A general extremal problem for the class of close-to-convex functionsHTML articles powered by AMS MathViewer

by John G. Milcetich
Trans. Amer. Math. Soc. 225 (1977), 307-323 Request permission

## Abstract:

For $\beta \geqslant 0,{K_\beta }$ denotes the set of functions $f(z) = z + {a_2}{z^2} + \cdots$ defined on the unit disc U with the representation $f’(z) = a{p^\beta }(z)s(z)/z$, where $a \in C$, p is an analytic function with positive real part in U, and s is a normalized starlike function. If $0 \leqslant \beta \leqslant 1$, and $\zeta \in U$, let $F(u,v)$ be analytic in a neighborhood of $\{ (f(\zeta ),\zeta ):f \in {K_\beta }\}$. Then $\max \{ \operatorname {Re} F(f(\zeta ),\zeta ):f \in {K_\beta }\}$ occurs for a function of the form $f(z) = {(\beta + 1)^{ - 1}}{(x - y)^{ - 1}}[{(1 + xz)^{\beta + 1}}{(1 + yz)^{ - \beta - 1}} - 1],$ where $|x| = |y| = 1$ and $x \ne y$. If $0 < \beta < 1$ these are the only extremal functions. A consequence of this result is the determination of the value region $\{ f(\zeta )/\zeta :f \in {K_\beta }\}$ as $\{ {(\beta + 1)^{ - 1}}{(s - t)^{ - 1}}[{(1 + s)^{\beta + 1}}{(1 + t)^{ - \beta - 1}} - 1]:|s|,|t| \leqslant |\zeta |\}$.
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