Asymptotic properties of meromorphic
solutions of differential equations in a neighborhood of a logarithmic singularity

L. I. Kolyasa; A. Z. Mokhonko; V. D. Mokhonko

We obtain asymptotic estimates of the moduli of meromorphic solutions with a logarithmic singularity at

$\infty$ of the differential equation

$\begin{gather*} \sum\limits_{k+s=m}f^kf_1^sv_{ks}(z)z^{\tau_{ks}}\text{Ln} ^{\varkappa_{ks}}z=\sum\limits_{|K|\leq m}b_K(z)f^{k_0}f_1^{k_1}\dots f_p^{k_p}, \\ f'=f_1, \ldots,f^{(p)}=f_p,\ K=(k_0,k_1,\ldots,k_p),\ |K|=k_0+k_1+\ldots+k_p;\\ \tau_{m-s,s}-s\leqslant\tau_{m-n,n}-n,\ s\leq n=\max\{s\colon k+s=m, c_{ks}\neq0\}; \end{gather*}$ where

$v_{ks}(z), b_K(z)$ are analytic functions such that

$\forall \alpha, \beta \in\mathbb{R},$

$\begin{array}{c} |b_K(re^{i\theta})|\leq r^{\tau_K},\ v_{ks}(re^{i\theta})=c_{ks}+o(1),\ r\to+\infty,\ \alpha\leqslant\theta\leqslant\beta;\ \tau_{ks},\ \varkappa_{ks},\ \tau_K\in\mathbb{R},\ c_{ks}\in\mathbb{C}. \end{array}$

1. Markushevich À.I. Theory of analytic functions. - Ì.: Nauka, 1967, V.1. 488p.; 1968, V.2. 624p. (in Russian)

2. Golubev V.V. Lectures on the analytic theory of differential equations. - Ì.L.: GITL, 1950. - 436p. (in Russian)

3. Boutroux P. Sur quelques properties des fonctions entieres// Acta math. 1904. V.29. P. 97204.

4. Van der Waerden B.L. Algebra. - Ì.: Nauka, 1979. - 624p. (in Russian)

5. Mokhonko A.Z., Kuzemko L.I. About logarithmic derivative of meromorphic function// Visnyk of Lviv Polytechnic National University, Physical and mathematical sciences. 2006. V.566. P. 1219. (in Ukrainian)

6. Goldberg À.À., Ostrovskii I.V. Value distribution of meromorphic functions. Ì.: Nauka, 1970. 592 p. (in Russian)

7. Mokhonko A.Z., Mokhonko V.D. On order of growth of analytic solutions for algebraic differential equations having logarithmic singularity// Mat. Stud. 2000. V.13, ¹2. P. 203218.

8. Mokhonko À.A., Mokhonko A.Z. On the logarithmic derivative of meromorphic functions// Topics in Analysis and its Applications. NATO Science Series. II Mathematics, Physics and Chemistry. 2004. V.147. P. 91103.

9. Mokhonko A.Z., Mokhonko V.D. Asymptotic estimates growth of meromorphic solutions of differential equations in an angular domain// Sib. Math. J. 2000. V.41, ¹1. P. 185199. (in Russian)

10. Mokhonko A.Z. Estimates of absolute value of the logarithmic derivative of the function meromorphic in an angular domain and its applications// Ukr. Mat. J. 1989. V.41, ¹6. P. 839843. (in Russian)

	Asymptotic properties of meromorphic solutions of differential equations in a neighborhood of a logarithmic singularity
Author	L. I. Kolyasa, A. Z. Mokhonko, V. D. Mokhonko kolyasa.lubov@gmail.com Lviv Polytechnic National University
Abstract	We obtain asymptotic estimates of the moduli of meromorphic solutions with a logarithmic singularity at $\infty$ of the differential equation $\begin{gather} \sum\limits_{k+s=m}f^kf_1^sv_{ks}(z)z^{\tau_{ks}}\text{Ln} ^{\varkappa_{ks}}z=\sum\limits_{\|K\|\leq m}b_K(z)f^{k_0}f_1^{k_1}\dots f_p^{k_p}, \\ f'=f_1, \ldots,f^{(p)}=f_p,\ K=(k_0,k_1,\ldots,k_p),\ \|K\|=k_0+k_1+\ldots+k_p;\\ \tau_{m-s,s}-s\leqslant\tau_{m-n,n}-n,\ s\leq n=\max\{s\colon k+s=m, c_{ks}\neq0\}; \end{gather}$ where $v_{ks}(z), b_K(z)$ are analytic functions such that $\forall \alpha, \beta \in\mathbb{R},$ $\begin{array}{c} \|b_K(re^{i\theta})\|\leq r^{\tau_K},\ v_{ks}(re^{i\theta})=c_{ks}+o(1),\ r\to+\infty,\ \alpha\leqslant\theta\leqslant\beta;\ \tau_{ks},\ \varkappa_{ks},\ \tau_K\in\mathbb{R},\ c_{ks}\in\mathbb{C}. \end{array}$
Keywords	meromorphic function; logarithmic singularity; differential equation
DOI	doi:10.30970/ms.42.1.67-83
Reference	1. Markushevich À.I. Theory of analytic functions. - Ì.: Nauka, 1967, V.1. 488p.; 1968, V.2. 624p. (in Russian) 2. Golubev V.V. Lectures on the analytic theory of differential equations. - Ì.L.: GITL, 1950. - 436p. (in Russian) 3. Boutroux P. Sur quelques properties des fonctions entieres// Acta math. 1904. V.29. P. 97204. 4. Van der Waerden B.L. Algebra. - Ì.: Nauka, 1979. - 624p. (in Russian) 5. Mokhonko A.Z., Kuzemko L.I. About logarithmic derivative of meromorphic function// Visnyk of Lviv Polytechnic National University, Physical and mathematical sciences. 2006. V.566. P. 1219. (in Ukrainian) 6. Goldberg À.À., Ostrovskii I.V. Value distribution of meromorphic functions. Ì.: Nauka, 1970. 592 p. (in Russian) 7. Mokhonko A.Z., Mokhonko V.D. On order of growth of analytic solutions for algebraic differential equations having logarithmic singularity// Mat. Stud. 2000. V.13, ¹2. P. 203218. 8. Mokhonko À.A., Mokhonko A.Z. On the logarithmic derivative of meromorphic functions// Topics in Analysis and its Applications. NATO Science Series. II Mathematics, Physics and Chemistry. 2004. V.147. P. 91103. 9. Mokhonko A.Z., Mokhonko V.D. Asymptotic estimates growth of meromorphic solutions of differential equations in an angular domain// Sib. Math. J. 2000. V.41, ¹1. P. 185199. (in Russian) 10. Mokhonko A.Z. Estimates of absolute value of the logarithmic derivative of the function meromorphic in an angular domain and its applications// Ukr. Mat. J. 1989. V.41, ¹6. P. 839843. (in Russian)
Pages	67-83
Volume	42
Issue	1
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Asymptotic properties of meromorphic solutions of differential equations in a neighborhood of a logarithmic singularity