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Univalency of certain analytic functions. (English) Zbl 1032.30010

Let \(\mathcal A\) be the class of functions that are analytic in the unit disc \({\mathcal U}=\{z:|z|<1\}\) and normalized such that \(f(0)=f'(0)-1=0.\) In this paper the authors introduce a subclass of \(\mathcal A,\) denoted by \({\mathcal T}(\lambda,\mu,g),\) and defined by: if \(g(z)\in {\mathcal A},\) with \(\frac{g(z)}{z}\neq 0,\) \(z\in{\mathcal U},\) \(\text{ Re} (\lambda) \geq 0\) and \(\mu>0\) then \(f(z)\in {\mathcal T}(\lambda,\mu,g)\) if and only if for all \(z\in {\mathcal U}\) \[ \frac{f(z)}{z}\neq 0 \] and \[ \left|z^2\left(\frac{f'(z)}{f^2(z)}-\frac{g'(z)}{g^2(z)}\right)-\lambda z^2\left(\frac{z}{f(z)}-\frac{z}{g(z)}\right)\right|<\mu. \] For this class the authors give a radius of univalence and necessary conditions for univalence and for that purpose they use some results from the theory of differential subordinations.

MSC:

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

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