Meromorphic functions with zeros and poles in small angles. II (Q923159)

This is a continuation of [Sib. Mat. Zh. 26, No. 4(152), 22--37 (1985; Zbl 0578.30017)], and freely uses notations from Chapter 6 of the text of \textit{A. A. Goldberg} and \textit{I. V. Ostrovskiĭ} [Distribution of values of meromorphic functions (Russian), Moscow: Nauka (1970; Zbl 0217.10002)], and the reader is forced to check these references. In his first paper, the author considers functions of order \(\rho <\infty\) almost all of whose zeros and poles lie in \(D\), where \(D\) is a finite union of sectors \(\{\alpha_j<\arg z<\beta_j\}\). In terms of the geometry of \(D\), he introduces various constants \(\omega(D)\), \(\omega'(D)\), \(\omega'_0(D)\). In addition, he lets \(\lambda_*\), \(\rho_*\) be the lower (upper) Pólya growth indices of \(f\) (note that \(\lambda_*\leq \rho \leq \rho_*)\), and takes \(\tau \in (\lambda_*,\rho_*)\). The author shows then that certain choices of \(\tau\) cannot hold, where the restrictions depend on the various omega's and non-zero deficiencies of \(f\). In the present paper, he shows that his conclusions apply in the weaker situation that \(\lambda_*<\infty\) (so that \(\rho\) may well be infinite). The proofs use standard potential theory and A. Baernstein's *-function.