О РАцИОНАльНых сОстА ВльУЩИх МЕРОМОРФНых ФУНкцИИ И Их пРОИжВОД Ных

Abstract

LetG be an arbitrary domain in$\bar C$,f a function meromorphic inG,

$$M_f \mathop = \limits^{def} \mathop {\lim \sup }\limits_{G \mathrel\backepsilon z \to \partial G} \left| {f(z)} \right|< \infty ,$$

andR the sum of the principal parts in the Laurent expansions off with respect to all its poles inG. We set

$$f_G (z) = R(z) - \alpha ,{\mathbf{ }}where{\mathbf{ }}\alpha = \mathop {\lim }\limits_{z \to \infty } (f(z) - R(z))$$

in case ∞∃G, andα=0 in case ∞∋G. It is proved that

$$\left\| {f_G } \right\|_{C(\partial G)} \leqq 50(\deg f_G )M_f ,{\mathbf{ }}\left\| {f'_G } \right\|_{L_1 (\partial G)} \leqq 50(\deg f_G )V(\partial G)M_f ,$$

where

$$V(\partial G) = \sup \left\{ {\left\| {r'} \right\|_{L_1 (\partial G)} :r(z) = a/(z - b),{\mathbf{ }}\left\| r \right\|_{G(\partial G)} \leqq 1} \right\}.$$