Distribution of zeros of certain meromorphic functions (Q913980)

The paper contains some misprints. To the reviewer's opinion the main result must be formulated in the following way. Theorem. Let B, D be two simply-connected domains in the complex plane \({\mathbb{C}}\), \(B\subseteq D\); \(\phi\) (z) be a finite nonvanishing function \({\mathbb{C}}\setminus B\to {\mathbb{C}}\); \(G_*(D')\) be the convex hull of the set \(\cup_{z\in {\mathbb{C}}\setminus D}G(z)\), where G(z) is the image of B under the mapping \(n\phi (z)/(z-\zeta)^ m\), \(\zeta\in B\), \(m\geq 1.\) Suppose \(z_ 1,...,z_ n\in B\); \(P_ n(z)=\prod^{n}_{j=1}(z-z_ j)\); \(p_ n(z)\) is a polynomial of degree \(\leq n\) which satisfies \(| p_ n| (z)| \leq | P_ n(z)|\) on the boundary \(\partial B\). Then the inequality \[ | P^ m_ n(z)| | \phi (z)\frac{d^ m\log p_ n(z)}{dz^ m}+(m-1)!w| \leq | P^ m_ n(z)| | \phi (z)\frac{d^ m\log P_ n(z)}{dz^ m}+(m- 1)!w| \] holds for every \(z\in {\mathbb{C}}\setminus D\), \(w\in {\mathbb{C}}\setminus G_*(D')\).