On the radius of starlikeness of certain analytic functions
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- by Hassoon S. Al-Amiri
- Proc. Amer. Math. Soc. 42 (1974), 466-474
- DOI: https://doi.org/10.1090/S0002-9939-1974-0330431-4
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Abstract:
Let $F(z)$ be regular in the unit disk $\Delta :|z| < 1$ and normalized by the conditions $F(0) = 0$ and $Fโ(0) = 1$. Let $f(z) = \tfrac {1}{2}[zF(z)]โ$. Recently Libera and Livingston have studied the mapping properties of $f(z)$ when $F(z)$ is known. In particular, they have determined the radius of starlikeness of order $\beta$ for $f(z)$ when $F(z)$ is starlike of order $\alpha ,0 \leqq \alpha \leqq \beta < 1$. The author extends this study to include the complementary case $0 \leqq \beta < \alpha$. Also, a different proof has been given to determine the disk in which $\operatorname {Re} \{ fโ(z)\} > \beta$ when $\operatorname {Re} \{ Fโ(z)\} > \alpha ,0 \leqq \alpha < 1,0 \leqq \beta < 1$. All results are sharp.References
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Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 466-474
- MSC: Primary 30A32
- DOI: https://doi.org/10.1090/S0002-9939-1974-0330431-4
- MathSciNet review: 0330431