Some applications of a first order differential subordination. (English) Zbl 1083.30016

Singh, Sukhjit; Gupta, Sushma

JIPAM, J. Inequal. Pure Appl. Math. 5, No. 3, Paper No. 78, 15 p. (2004).

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 \(E\).

Reviewer: Maslina Darus (Selangor)

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

Keywords:

univalent function; starlike function; convex function; differential subordination