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On the Fekete-Szegő problem for close-to-convex functions


Author: Wolfram Koepf
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
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Abstract: Let $ S$ be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result

$\displaystyle {\max _{f \in S}}{\text{\vert}}{a_3} - \lambda a_2^2\vert = 1 + 2{e^{ - 2\lambda /1 - \lambda )}}$

for $ \lambda \in [0,1]$. We consider the corresponding problem for the family $ C$ of close-to-convex functions and get

$\displaystyle \max\limits_{f \in C} \vert{a_3} - \lambda a_2^2 = \left\{ {\begi... ... & {{\text{if}}\lambda \in {\text{[2/3,1]}}{\text{.}}} \\ \end{array} } \right.$

As an application it is shown that $ \vert\vert{a_3}\vert - \vert{a_2}\vert\vert \leq 1$ for close-to-convex functions, in contrast to the result in $ S$

$\displaystyle \mathop {\max }\limits_{f \in s} {\text{\vert\vert}}{a_3}\vert - \vert{a_2}\vert\vert = 1.029....$


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0897076-8
Keywords: Close-to-convex functions, univalent functions
Article copyright: © Copyright 1987 American Mathematical Society

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