On the radius of starlikeness of certain analytic functions

Al-Amiri, Hassoon S.

doi:10.1090/S0002-9939-1974-0330431-4

On the radius of starlikeness of certain analytic functions
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by Hassoon S. Al-Amiri PDF

Proc. Amer. Math. Soc. 42 (1974), 466-474 Request permission

Abstract:

Let $F(z)$ be regular in the unit disk $\Delta :|z| < 1$ and normalized by the conditions $F(0) = 0$ and $F’(0) = 1$. Let $f(z) = \tfrac {1}{2}[zF(z)]’$. 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 $\operatorname {Re} \{ f’(z)\} > \beta$ when $\operatorname {Re} \{ F’(z)\} > \alpha ,0 \leqq \alpha < 1,0 \leqq \beta < 1$. All results are sharp.

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