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 А.I. Theory of analytic functions. - М.: Nauka, 1967, V.1. – 488p.; 1968, V.2. – 624p. (in Russian)

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

3. Boutroux P. Sur quelques properties des fonctions entieres// Acta math. – 1904. – V.29. – P. 97–204.

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

5. Mokhon’ko 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. 12–19. (in Ukrainian)

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

7. Mokhon’ko A.Z., Mokhon’ko V.D. On order of growth of analytic solutions for algebraic differential equations having logarithmic singularity// Mat. Stud. – 2000. – V.13, №2. – P. 203–218.

8. Mokhon’ko А.A., Mokhon’ko 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. 91–103.

9. Mokhon’ko A.Z., Mokhon’ko V.D. Asymptotic estimates growth of meromorphic solutions of differential equations in an angular domain// Sib. Math. J. – 2000. – V.41, №1. – P. 185–199. (in Russian)

10. Mokhon’ko 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, №6. – P. 839–843. (in Russian)

	Asymptotic properties of meromorphic solutions of differential equations in a neighborhood of a logarithmic singularity
Author	L. I. Kolyasa, A. Z. Mokhon’ko, V. D. Mokhon’ko 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
Reference	1. Markushevich А.I. Theory of analytic functions. - М.: Nauka, 1967, V.1. – 488p.; 1968, V.2. – 624p. (in Russian) 2. Golubev V.V. Lectures on the analytic theory of differential equations. - М.–L.: GITL, 1950. - 436p. (in Russian) 3. Boutroux P. Sur quelques properties des fonctions entieres// Acta math. – 1904. – V.29. – P. 97–204. 4. Van der Waerden B.L. Algebra. - М.: Nauka, 1979. - 624p. (in Russian) 5. Mokhon’ko 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. 12–19. (in Ukrainian) 6. Goldberg А.А., Ostrovskii I.V. Value distribution of meromorphic functions. – М.: Nauka, 1970. – 592 p. (in Russian) 7. Mokhon’ko A.Z., Mokhon’ko V.D. On order of growth of analytic solutions for algebraic differential equations having logarithmic singularity// Mat. Stud. – 2000. – V.13, №2. – P. 203–218. 8. Mokhon’ko А.A., Mokhon’ko 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. 91–103. 9. Mokhon’ko A.Z., Mokhon’ko V.D. Asymptotic estimates growth of meromorphic solutions of differential equations in an angular domain// Sib. Math. J. – 2000. – V.41, №1. – P. 185–199. (in Russian) 10. Mokhon’ko 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, №6. – P. 839–843. (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