## On the growth of the logarithmic derivative.(English)Zbl 1013.30018

Let $$f$$ be a meromorphic function on $$\mathbb{C}$$ with $$f(0)=1$$. A. Gol’dberg and V. Grinshtein [Math. Notes 19, 320-323 (1976; Zbl 0338.30020)] sharpened the lemma of logarithmic derivative in Nevanlinna theory as follows: $m\left(r,\frac{f'}{f}\right)\leq \log^+\frac{\rho T(\rho,f)}{r(\rho-r)}+ c,$ where $$\rho>r$$, $$c\leq 5.8501$$. In this paper, the author proved $$c\leq 5.3078$$.
Reviewer: Pei-Chu Hu (Jinan)

### MSC:

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

### Keywords:

meromorphic function; logarithmic derivative; growth; error term

Zbl 0338.30020
Full Text:

### References:

 [1] W. Cherry and Z. Ye, Nevanlinna’s Theory of Value Distribution. The Second Main Theorem and its Error Terms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. · Zbl 0981.30001 [2] A. Gol’dberg and V. Grinshtein, The logarithmic derivative of a meromorphic function (Russian), Mat. Zametki 19 (1976), 525–530; English translation: Math. Notes 19 (1976), 320–323. [3] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. [4] A. Hinkkanen, A sharp form of Nevanlinna’s second fundamental theorem, Invent. Math. 108 (1992), 549–574. · Zbl 0766.30025 [5] M. Jankowski, An estimate for the logarithmic derivative of meromorphic functions, Analysis 14 (1994), 185–194. · Zbl 0807.30018 [6] A. Kolokolnikov, On the logarithmic derivative of a meromorphic function (Russian), Mat. Zametki 15 (1974), 711–718. [7] S. Lang, Lectures on Nevanlinna theory, in S. Lang and W. Cherry, Topics in Nevanlinna Theory, Lecture Notes in Math, 1433, Springer-Verlag, 1990. · Zbl 0735.30032 [8] C. Osgood C, Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds or better, J. Number Theory 21 (1985), 347–398. · Zbl 0575.10032 [9] W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005 [10] V. Smirnov, Sur les valeurs limites des fonctions régulières à l’interieur d’un cercle, Journal Leningr. Fiz. Mat. 2 (1928), 22–37. [11] P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math, 1239, Springer-Verlag, 1987. · Zbl 0609.14011
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