Stability phenomenon and problems for complex differential equations with relations to shared values

G.A.Barsegian; I.Laine; C.C.Yang

Three results and a~collection of problems for complex algebraic differential equations and their systems related with so-called stability phenomenon are formulated. The latter, in particular, means that meromorphic (entire) solutions of algebraic differential equation $P(z,f,f',f, \dots,$ $f^{(k)})=0$, where $P(\cdot)$ is a~polynomial in all variables, retain properties which they have on a ``small'' subset of $\Bbb C$.

1. Barsegian, G. A. Estimates of derivatives of meromorphic functions on set of $a$-points, J. London Math. Soc. (2) 34 (1986), 534--540.

2. Barsegian, G. A. and Yang, C. C. Some new generalizations in the theory of meromorphic functions and their applications. Part 1. On the magnitudes of functions on the set of $a$-points of derivatives, To appear in Complex Variables Theory Appl.

3. Barsegian, G. A. and Yang, C. C. Some new generalizations in the theory of meromorphic functions and their applications. Part 2. On the derivatives of meromorphic and inverse functions, To appear in Complex Variables Theory Appl.

4. Barsegian, G. A., Laine, I. and Yang, C. C. On a method of estimating derivatives in complex differential equations, Submitted

5. Dobbertin, H. and Schmieder, G. Zur Charakterisierung von Polynomen durch ihre Null- und Einsstellen, Arch. Math. 48 (1987), 337–347. russian

6. Гольдберг А. А. Об однозначных интегралах дифференциальных уравнений первого порядка, Укр. мат. журн. 8 (1956), 254–261.

7. Hua, X. H. and Yang C. C. Uniqueness problems of entire and meromorphic functions, Bull. Hong Kong Math. Soc. 1 (1997), 289–300

8. Laine, I. Nevanlinna theory and complex differential equation, W. de Gruyter, Berlin–New York, 1993.

9. Li, K.-S. and Chan, W. Meromorphic solutions of higher order system of algebraic differential equations, Math. Scand. 71 (1992), 105–121.

10. Moh, T. T. On a certain group structure for polynomials, Proc. Amer. Math. Soc. 82 (1981), 183–187.

11. Mues, E. Shared value problems for meromorphic functions, In: Univ. of Joensuu Publ. Sci. 35 (1995), 17–43 .

12. Ostrovskii, I., Pakovitch, F. and Zaidenberg, M. A remark on complex polynomials of least deviation, Internat. Math. Res. Notices 14 (1996), 699–703.

13. Pakovitch, F. Sur un probléme d'unicité pour les fonctions méromophes, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 745–748.

14. Problems in complex function theory, In: Complex analysis (Proc. S.U.N.Y. Conf., Brockport, N.Y. 1976), 167–177. Lecture Notes in Pure and Appl. Math. 36, Dekker, New York (1978).

15. Yang, C. C. Unicity and factorization of meromorphic functions, In: Proc. of the Second Asian Math. Conf. 1995, World Sci. Publishing (1998), 64–84.

	Stability phenomenon and problems for complex differential equations with relations to shared values
Author	G.A.Barsegian, I.Laine, C.C.Yang Institute,of Mathematics, National Academy of Sciences of Armenia,Marshal Bagramian,ave, 24–b, Yerevan 375019, Armenia, Department of Mathematics, University of Joensuu, P.O. Box 111, FIN–80101 Joensuu, Finland, Department,of Mathematics, The Hong Kong University of Science and Technology,Clear,Water Bay, Kowloon, Hong Kong
Abstract	Three results and a~collection of problems for complex algebraic differential equations and their systems related with so-called stability phenomenon are formulated. The latter, in particular, means that meromorphic (entire) solutions of algebraic differential equation $P(z,f,f',f, \dots,$ $f^{(k)})=0$, where $P(\cdot)$ is a~polynomial in all variables, retain properties which they have on a ``small'' subset of $\Bbb C$.
Keywords	complex algebraic differential equations, meromorphic solutions, stability phenomenon, polynomial differential equations, systems of differential equations, complex analysis
DOI	doi:10.30970/ms.13.2.224-228
Reference	1. Barsegian, G. A. Estimates of derivatives of meromorphic functions on set of $a$-points, J. London Math. Soc. (2) 34 (1986), 534--540. 2. Barsegian, G. A. and Yang, C. C. Some new generalizations in the theory of meromorphic functions and their applications. Part 1. On the magnitudes of functions on the set of $a$-points of derivatives, To appear in Complex Variables Theory Appl. 3. Barsegian, G. A. and Yang, C. C. Some new generalizations in the theory of meromorphic functions and their applications. Part 2. On the derivatives of meromorphic and inverse functions, To appear in Complex Variables Theory Appl. 4. Barsegian, G. A., Laine, I. and Yang, C. C. On a method of estimating derivatives in complex differential equations, Submitted 5. Dobbertin, H. and Schmieder, G. Zur Charakterisierung von Polynomen durch ihre Null- und Einsstellen, Arch. Math. 48 (1987), 337–347. russian 6. Гольдберг А. А. Об однозначных интегралах дифференциальных уравнений первого порядка, Укр. мат. журн. 8 (1956), 254–261. 7. Hua, X. H. and Yang C. C. Uniqueness problems of entire and meromorphic functions, Bull. Hong Kong Math. Soc. 1 (1997), 289–300 8. Laine, I. Nevanlinna theory and complex differential equation, W. de Gruyter, Berlin–New York, 1993. 9. Li, K.-S. and Chan, W. Meromorphic solutions of higher order system of algebraic differential equations, Math. Scand. 71 (1992), 105–121. 10. Moh, T. T. On a certain group structure for polynomials, Proc. Amer. Math. Soc. 82 (1981), 183–187. 11. Mues, E. Shared value problems for meromorphic functions, In: Univ. of Joensuu Publ. Sci. 35 (1995), 17–43 . 12. Ostrovskii, I., Pakovitch, F. and Zaidenberg, M. A remark on complex polynomials of least deviation, Internat. Math. Res. Notices 14 (1996), 699–703. 13. Pakovitch, F. Sur un probléme d'unicité pour les fonctions méromophes, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 745–748. 14. Problems in complex function theory, In: Complex analysis (Proc. S.U.N.Y. Conf., Brockport, N.Y. 1976), 167–177. Lecture Notes in Pure and Appl. Math. 36, Dekker, New York (1978). 15. Yang, C. C. Unicity and factorization of meromorphic functions, In: Proc. of the Second Asian Math. Conf. 1995, World Sci. Publishing (1998), 64–84.
Pages	224-228
Volume	13
Issue	2
Year	2000
Journal	Matematychni Studii
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