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Steklov Institute of Mathematics at St.Petersburg

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PREPRINT 23/2005

M.F. Gamal'##
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.

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