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Some criteria for multivalently starlikeness. (English) Zbl 0972.30007
Nunokawa, Owa, and others have proved a number of theorems involving criteria for functions to be multivalently starlike. In this paper, the author obtains some improvements on these earlier results. For example, a corollary of the main theorem of the paper shows that if $f(z) = z + \sum_{k=2}^\infty a_k z^k$ is analytic in the unit disk with $f(z)f'(z) \ne 0$ there and satisfies Re $\{(1 + zf''(z)/f'(z))/(zf'(z)/f(z))\} < 1 + 1/(4 \log 2)$, then $|zf'(z)/f(z) - 1|< 1$ (and this result is sharp). The main theorem of the paper has a number of other corollaries and is of some independent interest: Let $\alpha > 0$, $p$, and $n$ be given. Suppose $g(z) = 1 + g_n z^n + \dots$ is analytic and non-zero in the unit disk. If Re $\{1 + \alpha z g'(z)/(p g^2(z))\} < M$ where $1 < M \le 1 + n\alpha/(2p \log 2)$, then Re $1/g(z) > 1 - 2p(M-1)\log 2/(n\alpha)$ in the disk. There are several other similar results.

30C45Special classes of univalent and multivalent functions
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