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New criteria for meromorphic univalent functions. (English) Zbl 0674.30014

Let \(M_ n\) denote the class of functions \(f(z)=z^{- 1}+\sum^{\infty}_{n=0}a_ nz^ n\) which are regular in \(0<| z| <1\) and satisfy the condition \[ Re[(D^{n+1}f)/(D^ nf)- 2]<(n/n+1),\quad for\quad | z| <1, \] where \[ D^ nf(z)=(1/z(1- z)^{n+1})*f(z),\quad n\geq 0 \] and * denotes the Hadamard product. In this paper the authors prove that \(M_{n+1}\subset M_ n\) further show that all functions in \(M_ n\) are univalent. The authors also consider certain integrals of functions in \(M_ n\).
Reviewer: R.M.Goel

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
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