Yong Chan KIM and Toshiyuki SUGAWA

*A conformal invariant for non-vanishing analytic functions and its applications*

### Abstract

The quantity $V_D(\varphi)=\sup_{z\in D}\rho_D(z)\inv
|\varphi'(z)/\varphi(z)|$ will be considered
for a non-vanishing analytic function $\varphi$ on
a plane domain $D$ with hyperbolic metric $\rho_D(z)|dz|.$
We see that this quantity has various nice properties such as
conformal invariance and monotoneity.
As a special case, for a proper subdomain $\Omega$
of the punctured plane $\C^*=\C\setminus\{0\},$
we define the domain constant $W(\Omega)=V_\Omega(\id),$
which will be called the circular width of $\Omega$ about the origin, and
we will see that $W(\Omega)$ dominates the value of $V_D(\varphi)$ if
$\varphi(D)\subset\Omega.$
As applications, we provide boundedness and univalence criteria for
those functions $f$ on the unit disk $\D$ for which $f'(\D)\subset\Omega.$
We also compute values of circular width for typical domains.

submission: 27 December, 2004

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