On the univalence of some classes of regular functions

Libera, R. J.; Livingston, A. E.

doi:10.1090/S0002-9939-1971-0288244-5

On the univalence of some classes of regular functions
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by R. J. Libera and A. E. Livingston PDF

Proc. Amer. Math. Soc. 30 (1971), 327-336 Request permission

Abstract:

Let $F(z)$ be regular in the unit disk $\Delta = \{ z:|z| < 1\}$ and normalized by the conditions $F(0) = 0$ and $F’(0) = 1$ and let $2f(z) = [zF(z)]’$. The paper deals with the mapping properties of $f(z)$ when $F(z)$ is known. For example, if $F(z)$ is starlike of order $\alpha , 0 \leq \alpha < 1$, then the disk in which $f(z)$ is always starlike of order $\beta , \alpha \leq \beta < 1$, is determined. All results are sharp.

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