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Transactions of the American Mathematical Society

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



Subordination-preserving integral operators

Authors: Sanford S. Miller, Petru T. Mocanu and Maxwell O. Reade
Journal: Trans. Amer. Math. Soc. 283 (1984), 605-615
MSC: Primary 30C80; Secondary 30C45
MathSciNet review: 737887
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Abstract: Let $ \beta $ and $ \gamma $ be complex numbers and let $ H$ be the space of functions regular in the unit disc. Subordination of functions $ f$, $ g \in H$ is denoted by $ f \prec g$. Let $ K \subset H$ and let the operator $ A:K \to H$ be defined by $ F = A(f)$, where

$\displaystyle F(z) = {\left[ {\frac{1} {{{z^\gamma }}}\int_0^z {{f^\beta }(t){t^{\gamma - 1}}dt} } \right]^{1/\beta }}.$

The authors determine conditions under which

$\displaystyle f \prec g \Rightarrow A(f) \prec A(g),$

and then use this result to obtain new distortion theorems for some classes of regular functions.

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Keywords: Subordination, integral operator, Loewner chain, univalent function
Article copyright: © Copyright 1984 American Mathematical Society

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