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.


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