Steklov Institute of Mathematics at St.Petersburg

PREPRINT 23/2005

A PROPERTY OF SOME MEROMORPHIC FUNCTION

This preprint was accepted December 23, 2005

ABSTRACT:

 We regard functions  $\Phi$ that are meromorphic in
 the unit disk $\Bbb D$ and that are the ratio of an inner function 
 and  a finite Blaschke product, where the degree of 
 an inner function is no less that degree of finite 
 Blaschke product. Let $\Omega_{\Phi}=\{z\in\Bbb D: |\Phi(z)|>1\}$.
Then the function $\Phi$ can be extended to a continuous function on 
$\operatorname{clos}\Omega_{\Phi}$ and the image under $\Phi$ of the
intersection of $\partial\Omega_{\Phi}$ and of the unit circle is
of zero Lebesgue measure  if and only if the
intersection of $\partial\Omega_{\Phi}$ and of the unit circle is
of zero Lebesgue measure.