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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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Distortions properties of alpha-starlike functions
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by Sanford S. Miller
Proc. Amer. Math. Soc. 38 (1973), 311-318
DOI: https://doi.org/10.1090/S0002-9939-1973-0310222-X

Abstract:

Let $\alpha$ be real and suppose that $f(z) = z + \Sigma _2^\infty {a_n}{z^n}$ is regular in the unit disc $D$ with $f(z)f’(z) \ne 0$ in $0 < |z| < 1$. If $\operatorname {Re} [(1 - \alpha )zf’(z)/f(z) + \alpha ((zf''(z)/f’(z)) + 1)] > 0$ for $z \in D$, then $f(z)$ is said to be an alpha-starlike function. These functions are univalent and they very naturally unify the classes of starlike $(\alpha = 0)$ and convex $(\alpha = 1)$ functions. The author obtains the $\tfrac {1}{4}$-theorem, sharp bounds on $|f(z)|$ and $|f’(z)|$, and growth conditions on $M(r)$.
References
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Bibliographic Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 38 (1973), 311-318
  • MSC: Primary 30A32
  • DOI: https://doi.org/10.1090/S0002-9939-1973-0310222-X
  • MathSciNet review: 0310222