On the transcendental entire solutions of a class of differential equations
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In this paper, we consider the differential equation F′ − Q1 = Reα(F − Q2), where Q1 and Q2 are polynomials with Q1Q2 6≡ 0,R is a rational function and α is an entire function. We consider solutions of the form F = fn, where f is an entire function and n ≥ 2 is an integer, and we prove that if f is a transcendental entire function, then Q1 Q2 is a polynomial and f′ = Q1 nQ2 f. This theorem improves some known results and answers an open question raised in [16].
Abstract
In this paper, we consider the differential equation F′ − Q1 = Reα(F − Q2), where Q1 and Q2 are polynomials with Q1Q2 6≡ 0,R is a rational function and α is an entire function. We consider solutions of the form F = fn, where f is an entire function and n ≥ 2 is an integer, and we prove that if f is a transcendental entire function, then Q1 Q2 is a polynomial and f′ = Q1 nQ2 f. This theorem improves some known results and answers an open question raised in [16].
참고문헌 (17)
- R. Br¨uck, "On entire functions which share one value CM with their first derivatives", Results Math, 30(1-2), pp.21~24 (1996)
- Z. X. Chen, "On conjecture of R. Br¨uck concerning the entire function sharing one value CM with its derivative", Taiwanese J. Math, 8(2), pp.235~244 (2004)
- J. Clunie, "On integral and meromorphic functions", J. London Math. Soc, 37, pp.17~22 (1962)
- W. Doeringer, "Exceptional value of differential polynomials", Pacific J. Math, 98(1), pp.55~52 (1982)
- G. G. Gundersen, "Entire functions that share one value with one or two of their derivatives", J. Math. Anal. Appl, 223(1), pp.88~95 (1998)
- W. K. Hayman, "Meromorphic Functions" (1964), Clarendon Press
- I. Laine, "Nevanlinna Theory and Complex Differential Equations" (1993), Walter de Gruyter
- C. L. Lei, "A note on unicity of meromorophic functions", Acta Math. Sci. Ser. A. Chin. Ed, 28(5), pp.802~807 (2008)
- Feng Lu, "A note on the Bruck conjecture", 대한수학회보, 48(5), pp.951~957 (2011), 대한수학회
- Feng Lu, "THE BRUCK CONJECTURE AND ENTIRE FUNCTIONS SHARING POLYNOMIALS WITH THEIR k-TH DERIVATIVES", 대한수학회지, 48(3), pp.499~512 (2011), 대한수학회
- E. Mues, "Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen", Manuscripta Math, 29(2-4), pp.195~206 (1979)
- L. A. Rubel, "Values shared by an entire function and its derivative, Complex Analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976)", Lecture Notes in Math., 599, pp.101~103 (1977), Springer
- L. Z. Yang, "Non-existence of meromorphic solutions of a Fermat type functional equation", Aequationes Math, 76(1-2), pp.140~150 (2008)
- H. X. Yi, "Uniqueness Theory of Meromorphic Functions" (1995), Science Press
- J. L. Zhang, "A power of an entire function sharing one value with its derivative", Comput. Math. Appl, 60(7), pp.2153~2160 (2010)
- J. L. Zhang, "A power of a meromorphic function sharing a small function with its derivative", Ann. Acad. Sci. Fenn. Math, 34(1), pp.249~260 (2009)
- T. D. Zhang, "Notes on a meromorphic function sharing one small function with its derivative", Complex Var. Elliptic Equ, 53(9), pp.857~867 (2008)