On some starlike and convex functions

Shah, G. M.

doi:10.1090/S0002-9947-1971-0269826-8

On some starlike and convex functions
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by G. M. Shah PDF

Trans. Amer. Math. Soc. 154 (1971), 83-91 Request permission

Abstract:

In this paper we study functions of the form $\smallint _0^z(g(t)/\Pi _{k = 1}^n{(1 - t{z_k})^{{\alpha _k}}})$ for $|z| < 1$ and show under what conditions such a function is convex, convex in one direction and hence univalent in $|z| < 1$. We also study the functions $g(z)$ where $g(0) = 1,g(z) \ne 0$ and $\operatorname {Re} \;[zg’(z)/g(z)] \geqq - \alpha ,0 \leqq \alpha < 1$, for $|z| < 1$.

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Beiträge zur theorie der Schlichten funktionen

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