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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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Regular functions $f(z)$ for which $zf^{โ€™} (z)$ is $\alpha$-spiral-like
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by Pran Nath Chichra
Proc. Amer. Math. Soc. 49 (1975), 151-160
DOI: https://doi.org/10.1090/S0002-9939-1975-0361033-2

Abstract:

Let $\mathfrak {F}_\alpha ^\lambda$ be the class of functions $f(z) = z + {a_2}{z^2} + \cdots$ which are regular in $E = \{ z/|z| < 1\}$ and satisfy \[ \operatorname {Re} \{ {e^{i\alpha }}(1 + zf''(z)/fโ€™(z))\} > \lambda \cos \alpha \] for some $\alpha ,|\alpha | < \pi /2$, and for some $\lambda ,0 \leq \lambda < 1$. The author finds a range on $\alpha$ for which $f(z)$ in $\mathfrak {F}_\alpha ^\lambda$ is univalent in $E$. In particular, the author improves upon the range on a for which $f(z) \in \mathfrak {F}_\alpha ^0$ is known to be univalent in $E$. Also a corresponding result is obtained for those functions $f(z)$ in $\mathfrak {F}_\alpha ^\lambda$ for which $f''(0) = 0$.
References
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Bibliographic Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 49 (1975), 151-160
  • MSC: Primary 30A32
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0361033-2
  • MathSciNet review: 0361033