On the logarithmic derivative of a meromorphic function

A. A. Kondratyuk; I. P. Kshanovskyy

We will prove that the inequality $$ m\left(r,\frac {f'}{f}\right)\le \log ^{+}\left(\frac {T(\rho,f)}{r}\frac {\rho}{\rho -r}\right)+4.8517, $$ where $\rho>r$, holds for all meromorphic functions such that $f(0)=1$. This is an improvement of the earlier results by Gol'dberg and Grinshtein, Benbourenane and Korhonen.

1. Гольдберг А.А., Гринштейн В. О логарифмической производной мероморфной функции, Мат. зам. 19 (1976), № 4, 525–530.

2. Benbourenane D., Korhonen R. On the growth of the logarithmic derivative, Computational Methods and Functional Theory. 1 (2001), № 2, 301–310.

3. Hayman W., Kennedy P. Subharmonic functions. V.1. Academic Press, London etc., 1976.

4. Hinkkanen A. Sharp error term in the Nevanlinna's theory , Complex differential and functional equations (Mekrijarvi) Univ. Joensuu Dept. Math., Rep. Ser. (2003), № 5, 51–79.

5. Jankowski M. An estimate for the logarihmic derivative of meromorphic functions, Analysis 14 (1994), 185–194.

	On the logarithmic derivative of a meromorphic function
Author	A. A. Kondratyuk, I. P. Kshanovskyy kshanovskyy@ukr.net Faculty of Mechanics and Mathematics ,Lviv Ivan Franko National University
Abstract	We will prove that the inequality $$ m\left(r,\frac {f'}{f}\right)\le \log ^{+}\left(\frac {T(\rho,f)}{r}\frac {\rho}{\rho -r}\right)+4.8517, $$ where $\rho>r$, holds for all meromorphic functions such that $f(0)=1$. This is an improvement of the earlier results by Gol'dberg and Grinshtein, Benbourenane and Korhonen.
Keywords	logarithmic derivative, meromorphic function, asymptotic estimate
DOI	doi:10.30970/ms.21.1.98-100
Reference	1. Гольдберг А.А., Гринштейн В. О логарифмической производной мероморфной функции, Мат. зам. 19 (1976), № 4, 525–530. 2. Benbourenane D., Korhonen R. On the growth of the logarithmic derivative, Computational Methods and Functional Theory. 1 (2001), № 2, 301–310. 3. Hayman W., Kennedy P. Subharmonic functions. V.1. Academic Press, London etc., 1976. 4. Hinkkanen A. Sharp error term in the Nevanlinna's theory , Complex differential and functional equations (Mekrijarvi) Univ. Joensuu Dept. Math., Rep. Ser. (2003), № 5, 51–79. 5. Jankowski M. An estimate for the logarihmic derivative of meromorphic functions, Analysis 14 (1994), 185–194.
Pages	98-100
Volume	21
Issue	1
Year	2004
Journal	Matematychni Studii
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