DIFFERENTIAL EQUATIONS RELATED TO FAMILY A

Title & Authors
DIFFERENTIAL EQUATIONS RELATED TO FAMILY A
Li, Ping; Meng, Yong;

Abstract
Let h be a meromorphic function with few poles and zeros. By Nevanlinna`s value distribution theory we prove some new properties on the polynomials in h with the coefficients being small functions of h. We prove that if f is a meromorphic function and if $\small{f^m}$ is identically a polynomial in h with the constant term not vanish identically, then f is a polynomial in h. As an application, we are able to find the entire solutions of the differential equation of the type \$\$f^n+P(f)
Keywords
Nevanlinna theory;meromorphic function;differential equation;
Language
English
Cited by
References
1.
G. Brosch, Eindeutigkeitssatze fur meromorphe Funktionen, Thesis, Technical University of Aachen, 1989.

2.
W. Doeringer, Exceptional values of differential polynomials, Pacific J. Math. 98 (1982), no. 1, 55-62.

3.
W. Hayman, Meromorphic Functions, Oxford, 1964.

4.
G. Jank and N. Terglane, Meromorphic functions sharing three values, Math. Pannon. 2 (1991), no. 2, 37-46.

5.
P. Li, Unicity of meromorphic functions in a certain family, Complex Var. Theory Appl. 48 (2003), no. 12, 1041-1054.

6.
P. Li and W.-J. Wang, Entire functions that share a small function with its derivative, J. Math. Anal. Appl. 328 (2007), no. 1, 743-751.

7.
P. Li and C.-C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J. 18 (1995), no. 3, 437-450.

8.
Y. Meng and P. Li, Functional equations related to family A, Complex Var. Elliptic Equ. 52 (2007), no. 7, 589-603.

9.
A. Mohon'ko, The Nevanlinna characteristics of certain meromorphic functions, Teor. Funkcii Funkcional. Anal. i Prilozen. No. 14 (1971), 83-87.

10.
G. Weissenborn, On the theorem of Tumura and Clunie, Bull. London Math. Soc. 18 (1986), no. 4, 371-373.

11.
C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, 557. Kluwer Academic Publishers Group, Dordrecht, 2003.