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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A general extremal problem for the class of close-to-convex functions


Author: John G. Milcetich
Journal: Trans. Amer. Math. Soc. 225 (1977), 307-323
MSC: Primary 30A32
MathSciNet review: 0507714
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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

$\displaystyle f(z) = {(\beta + 1)^{ - 1}}{(x - y)^{ - 1}}[{(1 + xz)^{\beta + 1}}{(1 + yz)^{ - \beta - 1}} - 1],$

where $ \vert x\vert = \vert y\vert = 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]:\vert s\vert,\vert t\vert \leqslant \vert\zeta \vert\} $.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0507714-5
PII: S 0002-9947(1977)0507714-5
Keywords: Close-to-convex functions, extremal problems, variational methods
Article copyright: © Copyright 1977 American Mathematical Society