A RESULT ON A CONJECTURE OF W. LÜ, Q. LI AND C. YANG

Title & Authors
A RESULT ON A CONJECTURE OF W. LÜ, Q. LI AND C. YANG
Majumder, Sujoy;

Abstract
In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let f be a transcendental entire function, n and k be two positive integers. If $\small{f^n-Q_1}$ and $\small{(f^n)^{(k)}-Q_2}$ share 0 CM, and $\small{n{\geq}k+1}$, then $\small{(f^n)^{(k)}{\equiv}{\frac{Q_2}{Q_1}}f^n}$. Furthermore, if \$Q_1
Keywords
meromorphic function;derivative;small function;
Language
English
Cited by
References
1.
R. Bruck, On entire functions which share one value CM with their first derivative, Results Math. 30 (1996), no. 1-2, 21-24.

2.
J. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17-22.

3.
G. G. Gundersen, Meromorphic functions that share finite values with their derivative, J. Math. Anal. Appl. 75 (1980), no. 2, 441-446.

4.
G. G. Gundersen and L. Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998), no. 1, 88-95.

5.
W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

6.
F. Lu and H. X. Yi, The Bruck conjecture and entire functions sharing polynomials with their k-th derivatives, J. Korean Math. Soc. 48 (2011), no. 3, 499-512.

7.
W. Lu, Q. Li, and C. Yang, On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289.

8.
E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer ersten und zweiten Ableitung einen endlichen Wert teilen, Complex Var. Theory Appl. 6 (1986), no. 1, 51-71.

9.
L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivative, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), pp. 101-103. Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977.

10.
C. C. Yang, On deficiencies of differential polynomials. II, Math. Z. 125 (1972), 107-112.

11.
L. Z. Yang, Entire functions that share finite values with their derivatives, Bull. Austral. Math. Soc. 41 (1990), no. 3, 337-342.

12.
L. Z. Yang and J. L. Zhang, Non-existence of meromorphic solutions of a Fermat type functional equation, Aequationes Math. 76 (2008), no. 1-2, 140-150.

13.
H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995.

14.
J. L. Zhang, Meromorphic functions sharing a small function with their derivatives, Kyungpook Math. J. 49 (2009), no. 1, 143-154.

15.
J. L. Zhang and L. Z. Yang, A power of a meromorphic function sharing a small function with its derivative, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 1, 249-260.

16.
J. L. Zhang and L. Z. Yang, A power of an entire function sharing one value with its derivative, Comput. Math. Appl. 60 (2010), no. 7, 2153-2160.