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New criteria for meromorphic \(p\)-valent starlike functions. (English) Zbl 0804.30012

The authors study the class, \(B_ n(\alpha)\), of functions of the form \[ f(z) = a_{-p} z^{-p} + \sum_{k=0}^{\infty} a_ kz^ k \quad (a_{-p} \neq 0,\;p \in N = \{1,2,\dots\} \] which are regular in the punctured unit disc \(\{z:0 < | z | < 1\}\) and satisfies the condition \[ \text{Re} \left\{ {D^{n + 1} f(z) \over D^ nf(z)} - (p+1) \right\} < - \alpha, \quad (n \in N \cup \{0\},\;0 \leq \alpha < p,\;| z |<1), \] where \(D^ nf(z) = a_{-p} z^{-p} + \sum_{k = 0}^{\infty} (p + k + 1)^ n a_ kz^ k\). Using the well-known lemma of Clunie-Jack, the authors of this paper proved the inclusion \(B_{n+1} (\alpha) \subset B_ (\alpha)\) and obtain some results concerning the Bernard integral transform. None of the results is sharp.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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