A NOTE ON THE BRÜCK CONJECTURE

Title & Authors
A NOTE ON THE BRÜCK CONJECTURE
Lu, Feng;

Abstract
In 1996, Br$\small{\ddot{u}}$ck studied the relation between f and f if an entire function f shares one value a CM with its first derivative f and posed the famous Br$\small{\ddot{u}}$ck conjecture. In this work, we generalize the value a in the Br$\small{\ddot{u}}$ck conjecture to a small function $\small{{\alpha}}$. Meanwhile, we prove that the Br$\small{\ddot{u}}$ck conjecture holds for a class of meromorphic functions.
Keywords
entire functions;Nevanlinna theory;uniqueness;normal family;
Language
English
Cited by
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ON THE TRANSCENDENTAL ENTIRE SOLUTIONS OF A CLASS OF DIFFERENTIAL EQUATIONS,;;;

대한수학회보, 2014. vol.51. 5, pp.1281-1289
1.
ON THE TRANSCENDENTAL ENTIRE SOLUTIONS OF A CLASS OF DIFFERENTIAL EQUATIONS, Bulletin of the Korean Mathematical Society, 2014, 51, 5, 1281
2.
Some results about functions that share functions with their derivative of higher order, Advances in Difference Equations, 2013, 2013, 1, 192
References
1.
R. Bruck, On entire functions which share one value CM with their first derivatives, Results Math. 30 (1996), no. 1-2, 21-24.

2.
J. M. Chang and Y. Z. Zhu, Entire functions that share a small function with their derivatives, J. Math. Anal. Appl. 351 (2008), 491-496.

3.
Z. X. Chen and K. H. Shon, On conjecture of R. Bruck concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math. 8 (2004), no. 2, 235-244.

4.
J. Grahl and C. Meng, Entire functions sharing a polynomial with their derivatives and normal families, Analysis. (Munich) 28 (2008), no. 1, 51-61.

5.
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), 88-95.

6.
W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

7.
X. J. Liu, S. Nevo, and X. C. Pang, On the kth derivative of meromorphic functions with zeros of multiplicity at least $-\kappa}+1$, J. Math. Anal. Appl. 348 (2008), no. 1, 516-529.

8.
F. Lu, J. F. Xu, and A. Chen, Entire functions sharing polynomials with their first derivatives, Arch. Math. (Basel) 92 (2009), no. 6, 593-601.

9.
F. Lu and H. X. Yi, On the uniqueness problems of meromorphic functions and their linear differential polynomials, J. Math. Anal. Appl. 362 (2010), no. 2, 301-312.

10.
E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Manuscripta Math. 29 (1979), no. 2-4, 195-206.

11.
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.

12.
J. Wang and H. X. Yi, Uniqueness theory of entire functions that share a small function with its differential polynomails, Indian J. Pure Appl. Math. 35 (2004), 1119-1129.

13.
C. C. Yang and H. X. Yi, The uniqueness theory of meromorphic functions, Mathematics and Its Applications, Science Press/Kluwer Acad. Publ, 2003.

14.
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.

15.
J. L. Zhang and L. Z. Yang, Some results related to a conjecture of R. Bruck concerning meromorphic functions sharing one small function with their derivatives, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 1, 141-149.

16.
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.