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Meromorphic functions with zeros and poles in small angles. II. (English. Russian original) Zbl 0711.30034

Sib. Math. J. 31, No. 2, 192-202 (1990); translation from Sib. Mat. Zh. 31, No. 2(180), 9-20 (1990).
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 A. A. Goldberg and 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.

MSC:

30D30 Meromorphic functions of one complex variable (general theory)
30D40 Cluster sets, prime ends, boundary behavior
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References:

[1] E. V. Gleizer, ?On meromorphic functions with zeros and poles in small angles,? Sib. Mat. Zh.,26, No. 4, 22-37 (1985). · Zbl 0661.60056 · doi:10.1007/BF00968958
[2] A. A. Gol’dberg and I. V. Ostrovskii, The Distribution of the Values of Meromorphic Functions [in Russian], Nauka, Moscow (1970).
[3] J. Clunie, ?On integral functions having prescribed asymptotic growth,? Can. J. Math.,17, No. 3, 396-404 (1966). · Zbl 0134.29103 · doi:10.4153/CJM-1965-040-8
[4] A. Baernstein II, ?Integral means, univalent functions and circular symmetrization,? Acta Math.,133, 139-169 (1975). · Zbl 0315.30021
[5] M. L. Sodin, ?Certain results on the growth of meromorphic functions of finite lower order,? in: Mathematical Physics, Functional Analysis-Collection of Scientific Works [in Russian], Naukova Dumka, Kiev (1986), pp. 102-113.
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