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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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A general extremal problem for the class of close-to-convex functions
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by John G. Milcetich PDF
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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 225 (1977), 307-323
  • MSC: Primary 30A32
  • DOI: https://doi.org/10.1090/S0002-9947-1977-0507714-5
  • MathSciNet review: 0507714