Concentration of zeros and poles, $h$-measures, and uniform logarithmic derivative estimates of meromorphic functions

I.E.Chyzhykov

We prove new uniform estimates of the logarithmic derivative $f'(re^{i\theta})/f(re^{i\theta})$ of a meromorphic function $f$ in terms of the Nevanlinna characteristic $T(r,f)$ and a number of poles and zeros in some neighborhood of $z$ outside an exceptional set of finite $h$-measure. Obtained results are sharp when $f$ has finite order of the growth and improve known results in the general case.

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

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

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

7. Chiang Y.-M., Feng S.-J. On the growth of logarithmic diffrences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. (to appear)

8. Macintyre A.J. Wiman's method and the `flat regions' of integral functions, Quart. J. Math. 9 (1938), 81--88.

9. 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.

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

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

12. Jankowski M. An estimate for the logarithmic derivative of meromorphic functions, Analysis 14 (1994), 185--194.

13. Chyzhykov I.E. Sharp logarithmic derivative estimates for meromorphic functions, Matem. Studii 27 (2007), no.2, 120--138.

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

15. 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.

16. 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.

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

18. Salo T.M., Skaskiv O.B., Trakalo O.M. On the best possible description of exeptional set in Wiman-Valiron theory for entire function, Matematychni Studii 16 (2001), № 2, 131--140.

	Concentration of zeros and poles, $h$-measures, and uniform logarithmic derivative estimates of meromorphic functions
Author	I.E.Chyzhykov ichyzh@lviv.farlep.net Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv,
Abstract	We prove new uniform estimates of the logarithmic derivative $f'(re^{i\theta})/f(re^{i\theta})$ of a meromorphic function $f$ in terms of the Nevanlinna characteristic $T(r,f)$ and a number of poles and zeros in some neighborhood of $z$ outside an exceptional set of finite $h$-measure. Obtained results are sharp when $f$ has finite order of the growth and improve known results in the general case.
Keywords	concentration of zeros, concentration of poles, h-measure, meromorphic function. logarithmic derivative estimate
DOI	doi:10.30970/ms.29.2.151-164
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. Gundersen G. Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988) 88-104. 5. Chyzhykov I., Gundersen G.G., Heittokangas J. Linear differential equations and logarithmic derivative estimates, Proc. London Math. Soc. (3) 86 (2003), 735--754. 6. Heittokangas J. On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss. 122 (2000) 1-54. 7. Chiang Y.-M., Feng S.-J. On the growth of logarithmic diffrences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. (to appear) 8. Macintyre A.J. Wiman's method and the `flat regions' of integral functions, Quart. J. Math. 9 (1938), 81--88. 9. 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. 10. Heittokangas J., Korhonen R., Rättyä J., Generalized logarithmic derivative estimates of Gol'dberg-Grinshtein type, Bull. London Math. Soc. 36 (2004), 105--114. 11. Miles J. A sharp form of the lemma of the logarithmic derivative, J. London Math. Soc. 45 (1992), 243--254. 12. Jankowski M. An estimate for the logarithmic derivative of meromorphic functions, Analysis 14 (1994), 185--194. 13. Chyzhykov I.E. Sharp logarithmic derivative estimates for meromorphic functions, Matem. Studii 27 (2007), no.2, 120--138. 14. Strelitz Sh. On upper bounds for the logarithmic derivative of a meromorphic function, Complex variables, 23 (1993), 131-143. 15. 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. 16. 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. 17. Levin B. Ja. Distribution of zeros of entire functions, revised edition, Transl. Math. Monographs, Volume 5, translated by R. P. Boas et al (Amer. Math. Soc., Providence, 1980). 18. Salo T.M., Skaskiv O.B., Trakalo O.M. On the best possible description of exeptional set in Wiman-Valiron theory for entire function, Matematychni Studii 16 (2001), № 2, 131--140.
Pages	151-164
Volume	29
Issue	2
Year	2008
Journal	Matematychni Studii
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