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Some applications of a first order differential subordination. (English) Zbl 1083.30016
Let \(p\) and \(q\) be analytic functions in the unit disc \(E={z:| z| <1}\), with \(p(0)=q(0)=1\). Assume that \(\alpha\) and \(\delta\) are real numbers such that \(0<\delta\leq 1\), \(\alpha+\delta\geq 0\). Let \(\beta\) and \(\gamma\) be complex numbers with \(\beta\neq 0\). The authors investigate the differential subordination \[ p^{\alpha}(z)\left(p(z)+\dfrac{zp'(z)}{\beta p(z)+\gamma}\right)^{\delta}\prec q^{\alpha}(z)\left(q(z)+\dfrac{zq'(z)}{\beta q(z)+\gamma}\right)^{\delta} \] for \(z\in E\), and obtained several sufficient conditions for starlikeness and univalence of functions analytic in the unit disc \(E\).

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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