## 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.)

### Keywords:

Loewner chain; quasiconformal mapping; univalent function
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