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Spread of functions meromorphic in the disc. (English. Russian original) Zbl 0749.30019

J. Sov. Math. 59, No. 1, 658-665 (1992); translation from Teor. Funkts., Funkts. Anal. Prilozh. 55, 104-113 (1991).
Let \(f\) be a function meromorphic in the unit disc. The paper contains estimations from below for the upper limits (as \(r\to 1)\) of the following quantities: \[ (1-r)^{-1}\text{ mes}(t:\log| f(re^{it})|>qT(r,f)/(1-r)), \]
\[ (1-r)^{-1}\text{ mes}(t:\log| f(re^{it})|>qT(r,f)), \]
\[ (1-r)^{-1}\text{ mes}(t:\log| f(re^{it})|>q \log M(r,f)), \] where \(0\leq q<\infty\); \(T\) and \(M\) are standard notations of the Nevanlinna theory.
These estimations are obtained in terms of the lower order of the function \(f\) and the quantities \(\delta(\infty,f)\), \(\beta(\infty,f)\), \(\hat\beta(\infty,f)\), where \(\delta\) is the Nevanlinna deficiency and the \(\beta\)’s are the Petrenko deviations [V. P. Petrenko, Entire curves (Russian) (1984; Zbl 0591.30030)].

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Citations:

Zbl 0591.30030
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References:

[1] R. Nevanlinna, Single-Valued Analytic Functions [in Russian], Moscow (1941).
[2] A. A. Gol’dberg and I. V. Ostrovskii, Distribution of the Values of Meromorphic Functions [in Russian], Moscow (1970).
[3] A. Edrei, ?Sums of deficiencies of meromorphic functions,? J. Anal. Math.,14, 79?104 (1970). · Zbl 0154.07402 · doi:10.1007/BF02806380
[4] A. Baernstein, ?Proof of Edrei’s conjecture,? Proc. London Math. Soc.,26, 418?434 (1973). · Zbl 0263.30024 · doi:10.1112/plms/s3-26.3.418
[5] J. M. Anderson and A. Baernstein, ?The size of the set on which a meromorphic function is large,? Proc. London Math. Soc.,35, 518?539 (1978). · Zbl 0381.30014 · doi:10.1112/plms/s3-36.3.518
[6] V. P. Petrenko, Entire Curves [in Russian], Khar’kov (1984). · Zbl 0591.30030
[7] I. I. Marchenko, ?The growth of meromorphic functions of finite lower order,? Dokl. Akad. Nauk SSSR,264, No. 5, 1077?1080 (1982).
[8] A. V. Krytov, ?The growth of meromorphic functions and analytic curves in the unit disc,? Rept. No. 1657-81, Deposited in VINITI July 20, 1981.
[9] I. I. Marchenko and A. I. Shcherba, ?The sizes of deviations of meromorphic functions,? Mat. Sborn., 53?60 (1990). · Zbl 0697.30033
[10] V. I. Krutin’, ?The sizes of Nevanlinna deficiencies for functions meromorphic when |z|&lt;1,? Izv. Akad. Nauk Arm. SSR, Ser. Mat.,8, No. 5, 347?358 (1973).
[11] I. I. Marchenko and A. I. Shcherba, ?The growth of functions of finite lower order that are meromorphic in the disc,? Dokl. Akad. Nauk SSSR,295, No. 4, 805?808 (1987). · Zbl 0648.30025
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