Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Fekete-Szegő problem for close-to-convex functions
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by Wolfram Koepf
Proc. Amer. Math. Soc. 101 (1987), 89-95
DOI: https://doi.org/10.1090/S0002-9939-1987-0897076-8

Abstract:

Let $S$ be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result \[ {\max _{f \in S}}{\text {|}}{a_3} - \lambda a_2^2| = 1 + 2{e^{ - 2\lambda /1 - \lambda )}}\] for $\lambda \in [0,1]$. We consider the corresponding problem for the family $C$ of close-to-convex functions and get \[ \max \limits _{f \in C} |{a_3} - \lambda a_2^2 = \left \{ {\begin {array}{*{20}{c}} {3 - 4\lambda } & {{\text {if}}\lambda \in [0,1/3],} \\ {1/3 + 4/(9\lambda )} & {{\text {if}}\lambda \in {\text {[1/3,2/3],}}} \\ 1 & {{\text {if}}\lambda \in {\text {[2/3,1]}}{\text {.}}} \\ \end {array} } \right .\] As an application it is shown that $||{a_3}| - |{a_2}|| \leq 1$ for close-to-convex functions, in contrast to the result in $S$ \[ \max \limits _{f \in s} {\text {||}}{a_3}| - |{a_2}|| = 1.029....\]
References
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Bibliographic Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 101 (1987), 89-95
  • MSC: Primary 30C45
  • DOI: https://doi.org/10.1090/S0002-9939-1987-0897076-8
  • MathSciNet review: 897076