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Differential subordination and Bazilevič functions. (English) Zbl 0859.30024

Let $p$, $\lambda$, $h$ and $\phi$ be analytic functions in the unit disc ${\Delta }$. In this paper, the author proves that, under suitable assumptions on the above functions, the following relation holds: $p\left(z\right)+\lambda \left(z\right)z{p}^{\text{'}}\left(z\right)\prec h\left(z\right)$ implies $p\left(z\right)\prec \phi \left(z\right)\prec h\left(z\right)$ for $z\in {\Delta }$, where $\prec$ denotes the subordination of functions.

Basing himself on this result he proves a sufficient condition for an analytic function to be starlike in ${\Delta }$. These results are some extensions of the classical result of Sakaguchi and Libera.

##### MSC:
 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable) 30C45 Special classes of univalent and multivalent functions
##### Keywords:
Bazilevič functions; Sakaguchi; Libera
##### References:
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