Sharp logarithmic derivative estimates for meromorphic functions

I.E.Chyzhykov

Using recent estimates of the logarithmic derivative of a polynomial due to J.M.Anderson and V.Ya.Eiderman, we prove sharp estimates of the logarithmic derivative $f'(re^{i\theta})/f(re^{i\theta})$ of a meromorphic function $f$ in the plane or in the unit disk outside sets of finite logarithmic measures of values $r$.

1. Гольдберг А.А., Островский И.В. Распределение значений мероморфных функций, М.Наука, 1970, 592 с.

2. Hayman W. K. Meromorphic functions, Clarendon Press, Oxford, 1964.

3. Стрелиц Ш. Асимптотические свойства аналитических решений дифференциальных уравнений. -- Вильнюс, Минтис, 1972, 468 с.

4. Гольдберг А.А., Гринштейн В.А. О логарифмической производной мероморфной функции, Мат.заметки 19 (1976), no.4, 525--530. Math.Notes 19 (1976), 320--323.

5. Miles J. A sharp form of the lemma of the logarithmic derivative, J. London Math. Soc. 45 (1992), 243--254.

6. Heittokangas J., Korhonen R., RÄTTYÄ J., Generalized logarithmic derivative estimates of Gol'dberg-Grinshtein type, Bull. London Math. Soc. 36 (2004), 105--114.

7. Gundersen G. Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988) 88--104.

8. Chyzhykov I., Gundersen G.G., Heittokangas J. Linear differential equations and logarithmic derivative estimates, Proc. London Math. Soc. (3) 86 (2003), 735--754.

9. Heittokangas J. On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss. 122 (2000) 1--54.

10. Strelitz Sh. On upper bounds for the logarithmic derivative of a meromorphic function, Complex variables, 23 (1993), 131--143.

11. Anderson J.M., Eiderman V.Ya. Estimates for the Cauchy transforms of point masses (the logarithmic derivative of a polynomial), Doklady Ross Akad. Nauk 401 (2005), no.5, 583--586.

12. Anderson J.M., Eiderman V.Ya. Cauchy transforms of point masses (the logarithmic derivative of polynomials ), Annals of Math. 163 (2006), no.3, 1057--1076.

13. Levin B. Ja. Distribution of zeros of entire functions, revised edition, Transl. Math. Monographs, V.5 (translated by R. P. Boas) et al (Amer. Math. Soc., Providence, 1980).

14. Сумик О.М., Чижиков І.Е. Мероморфні функції та лінійні диференціальні рівняння, Львів, Львівський нац. ун-т, 2005. -- 98 с.

	Sharp logarithmic derivative estimates for meromorphic functions
Author	I.E.Chyzhykov ichyzh@lviv.farlep.net Faculty of Mechanics and Mathematics, Lviv Ivan Franko National University,
Abstract	Using recent estimates of the logarithmic derivative of a polynomial due to J.M.Anderson and V.Ya.Eiderman, we prove sharp estimates of the logarithmic derivative $f'(re^{i\theta})/f(re^{i\theta})$ of a meromorphic function $f$ in the plane or in the unit disk outside sets of finite logarithmic measures of values $r$.
Keywords	logarithmic derivative, sharp estimate, meromorphic function, unit disk, finite logarithmic measure
DOI	doi:10.30970/ms.27.2.120-138
Reference	1. Гольдберг А.А., Островский И.В. Распределение значений мероморфных функций, М.Наука, 1970, 592 с. 2. Hayman W. K. Meromorphic functions, Clarendon Press, Oxford, 1964. 3. Стрелиц Ш. Асимптотические свойства аналитических решений дифференциальных уравнений. -- Вильнюс, Минтис, 1972, 468 с. 4. Гольдберг А.А., Гринштейн В.А. О логарифмической производной мероморфной функции, Мат.заметки 19 (1976), no.4, 525--530. Math.Notes 19 (1976), 320--323. 5. Miles J. A sharp form of the lemma of the logarithmic derivative, J. London Math. Soc. 45 (1992), 243--254. 6. Heittokangas J., Korhonen R., RÄTTYÄ J., Generalized logarithmic derivative estimates of Gol'dberg-Grinshtein type, Bull. London Math. Soc. 36 (2004), 105--114. 7. Gundersen G. Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988) 88--104. 8. Chyzhykov I., Gundersen G.G., Heittokangas J. Linear differential equations and logarithmic derivative estimates, Proc. London Math. Soc. (3) 86 (2003), 735--754. 9. Heittokangas J. On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss. 122 (2000) 1--54. 10. Strelitz Sh. On upper bounds for the logarithmic derivative of a meromorphic function, Complex variables, 23 (1993), 131--143. 11. Anderson J.M., Eiderman V.Ya. Estimates for the Cauchy transforms of point masses (the logarithmic derivative of a polynomial), Doklady Ross Akad. Nauk 401 (2005), no.5, 583--586. 12. Anderson J.M., Eiderman V.Ya. Cauchy transforms of point masses (the logarithmic derivative of polynomials ), Annals of Math. 163 (2006), no.3, 1057--1076. 13. Levin B. Ja. Distribution of zeros of entire functions, revised edition, Transl. Math. Monographs, V.5 (translated by R. P. Boas) et al (Amer. Math. Soc., Providence, 1980). 14. Сумик О.М., Чижиков І.Е. Мероморфні функції та лінійні диференціальні рівняння, Львів, Львівський нац. ун-т, 2005. -- 98 с.
Pages	120-138
Volume	27
Issue	2
Year	2007
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html