## 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 $$f(z)f'(z) \neq 0$$ 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 $$1 < M \leq 1 + n\alpha/(2p \log 2)$$, then Re $$1/g(z) > 1 - 2p(M-1)\log 2/(n\alpha)$$ in the disk. There are several other similar results.

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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