Abstract
The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation \(f' = \frac{{P(z,f)}}{{Q(z,f)}}\), where P(z, f) and Q(z, f) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity.
Similar content being viewed by others
REFERENCES
A. A. Gol'dberg and I. V. Ostrovskii, Distribution of Values of Meromorphic Functions [in Russian], Nauka, Moscow (1970).
J. Malmquist, “Sur les fonctions a un nombre fini de branches definies par les equations differentielles du premier ordre,” Acta Math., 36, 297–343 (1913).
V. V. Golubev, Lectures on the Analytic Theory of Differential Equations [in Russian], Gostekhteorizdat, Moscow (1950).
K. Yosida, “A generalization of a Malmquist's theorem,” Jpn. J. Math., 9, 239–256 (1933).
A. A. Gol'dberg, B. Ya. Levin, and I. V. Ostrovskii, “Entire and meromorphic functions,” in: VINITI Series in Contemporary Problems of Mathematics, Fundamental Trends [in Russian], Vol. 85, VINITI, Moscow (1991), pp. 5–186.
I. Laine, Nevanlinna Theory and Complex Differential Equations, Gruyter, Berlin (1993).
B. L. van der Waerden, Algebra [Russian translation], Nauka, Moscow (1979).
A. Z. Mokhon'ko, “Malmquist theorem for solutions of differential equations in a neighborhood of a logarithmic singular point,” Ukr. Mat. Zh., 56, No.4, 476–483 (2004).
G. Valiron, “Sur la derivee des fonctions algebroides,” Bull. Soc. Math. France, 59, No.1–2, 17–39 (1931).
A. I. Markushevich, Theory of Analytic Functions [in Russian], Vol. 2, Nauka, Moscow (1968).
A. Z. Mokhon'ko, “A field of algebroidal functions and estimates for their Nevanlinna characteristics,” Sib. Mat. Zh., 22, No.3, 214–218 (1981).
A. Z. Mokhon'ko and V. D. Mokhon'ko, “On order of growth of analytic solutions for algebraic differential equations having logarithmic singularity,” Mat. Stud., 13, No.2, 203–218 (2000).
A. Z. Mokhon'ko, “On meromorphic solutions of algebraic differential equations in angular domains,” Ukr. Mat. Zh., 44, No.4, 514–523 (1992).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 505–513, April, 2005.
Rights and permissions
About this article
Cite this article
Mokhon'ko, A.A. On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point. Ukr Math J 57, 610–620 (2005). https://doi.org/10.1007/s11253-005-0214-7
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-005-0214-7