Entire Functions and Their Derivatives Share Two Finite Sets

  • Meng, Chao (Department of Mathematics, Shandong University) ;
  • Hu, Pei-Chu (Department of Mathematics, Shandong University)
  • Received : 2008.03.23
  • Accepted : 2008.05.16
  • Published : 2009.09.30


In this paper, we study the uniqueness of entire functions and prove the following theorem. Let n(${\geq}$ 5), k be positive integers, and let $S_1$ = {z : $z^n$ = 1}, $S_2$ = {$a_1$, $a_2$, ${\cdots}$, $a_m$}, where $a_1$, $a_2$, ${\cdots}$, $a_m$ are distinct nonzero constants. If two non-constant entire functions f and g satisfy $E_f(S_1,2)$ = $E_g(S_1,2)$ and $E_{f^{(k)}}(S_2,{\infty})$ = $E_{g^{(k)}}(S_2,{\infty})$, then one of the following cases must occur: (1) f = tg, {$a_1$, $a_2$, ${\cdots}$, $a_m$} = t{$a_1$, $a_2$, ${\cdots}$, $a_m$}, where t is a constant satisfying $t^n$ = 1; (2) f(z) = $de^{cz}$, g(z) = $\frac{t}{d}e^{-cz}$, {$a_1$, $a_2$, ${\cdots}$, $a_m$} = $(-1)^kc^{2k}t\{\frac{1}{a_1},{\cdots},\frac{1}{a_m}\}$, where t, c, d are nonzero constants and $t^n$ = 1. The results in this paper improve the result given by Fang (M.L. Fang, Entire functions and their derivatives share two finite sets, Bull. Malaysian Math. Sc. Soc. 24(2001), 7-16).


entire function;share set;uniqueness


  1. A. Banerjee, On uniqueness of meromorphic functions when two differential monomials share one value, Bull. Korean Math. Soc., 44(2007), 607-622.
  2. A. Banerjee, Meromorphic functions sharing one value, Int. J. Math. Math. Sci., 22(2005), 3587-3598.
  3. M. L. Fang, Entire functions and their derivatives share two finite sets, Bull. Malaysian Math. Sc. Soc., 24(2001), 7-16.
  4. F. Gross, Factorization of meromorphic functions and some open problems, Lecture Notes in Math. 599, Springer, Berlin, 1977.
  5. W. K. Hayman, Meromorphic Functions, Clarendon, Oxford, 1964.
  6. I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161(2001), 193-206.
  7. I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Variables Theory Appl., 46(2001), 241-253.
  8. C. C. Yang, On deficiencies of di erential polynomials, Math. Z., 125(1972), 107-112.
  9. L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993.
  10. H. X. Yi, On the uniqueness of meromorphic functions and a problem of Gross, Science in China, 24(1994), 457-466.
  11. H. X. Yi, Meromorphic functions that share one or two values, Complex Variables Theory Appl., 28(1995), 1-11.