## 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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