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Ruscheweyh’s univalence criterion and quasiconformal extensions. (English) Zbl 1208.30023

Let \(\mathcal A\) be the family of analytic functions \(f(z)=z+\sum_{n=2}^{\infty}a_nz^n\) on the unit disk \(\mathbb D\). The purpose of the paper is to refine Ruscheweyh’s univalence condition to a quasiconformal extension criterion proved in
Theorem 1: Let \(s=a+ib\), \(a>0\), \(b\in\mathbb R\), \(k\in[0,1)\), and \(f\in\mathcal A\). Assume that for some constant \(c\in\mathbb C\) and all \(z\in\mathbb D\),
\[ \left|c|z|^2+s-a\big(1-|z|^2\big)\left[s\left(1+\frac{zf''(z)}{f'(z)}\right)+(1-s)\frac{zf'(z)}{f(z)}\right]\right|\leq M \] with \(M=ak|s|+(a-1)|s+c|\) if \(0<a\leq1\) and \(M=k|s|\) if \(a>1\). Then \(f\) has an \(l\)-quasiconformal extension to \(\mathbb C\), where \[ l=\frac{2ka+(1-k^2)|b|}{(1+k^2)a+(1-k^2)|s|}<1. \] Theorem 1 includes Becker’s univalence condition. The author applies Theorem 1 to obtain a quasiconformal extension criterion for the class of Bazilevič functions. Besides, the author proves a similar result for functions \(g(\zeta)\) that are analytic on the exterior domain of \(\mathbb D\). This yields a corollary which gives a positive answer to an open problem posed by Ruscheweyh, whether a function \[ g(\zeta)=\zeta+\frac{d}{\zeta}+\dots,\qquad |\zeta|>1, \]
with
\[ \big(|\zeta|^2-1\big)\bigg|1+\frac{\zeta f''(\zeta)}{f'(\zeta)}-\frac{\zeta f'(\zeta)}{f(\zeta)}\bigg|\leq k\quad\text{for all}\quad |\zeta|>1 \] admits a quasiconformal extension to \(\mathbb C\).

MSC:

30C62 Quasiconformal mappings in the complex plane
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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