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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 \(f(z)f'(z) \neq 0\) 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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