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Transactions of the American Mathematical Society

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

 

 

Majorization-subordination theorems for locally univalent functions. III


Author: Douglas Michael Campbell
Journal: Trans. Amer. Math. Soc. 198 (1974), 297-306
MSC: Primary 30A42
MathSciNet review: 0349987
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Abstract: A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to $ {\mathfrak{n}_\alpha }$, the linear invariant family of locally univalent functions of finite order $ \alpha $. If $ f(z)$ is subordinate to $ F(z)$ in the open unit disc, $ f'(0) \geqslant 0$, and $ F(z)$ is in $ {\mathfrak{n}_\alpha },1.65 \leqslant \alpha < \infty $, then $ f'(z)$ is majorized by $ F'(z)$ in $ \vert z\vert \leqslant (\alpha + 1) - {({\alpha ^2} + 2\alpha )^{1/2}}$. The result is sharp.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0349987-5
Keywords: Linear invariant family, univalent analytic function, majorization, order of a linear invariant family, subordination
Article copyright: © Copyright 1974 American Mathematical Society