[1] Zhang Minzhu,. On the uniqueness of entire functions [J]. Journal of Southeast University (English Edition), 2004, 20 (3): 387-394. [doi:10.3969/j.issn.1003-7985.2004.03.025]

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On the uniqueness of entire functions()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

20

Issue:

2004 3

Page:

387-394

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2004-09-30

Info

Title:

On the uniqueness of entire functions

Author(s):

Zhang Minzhu

Department of Mathematics, Southeast University, Nanjing 210096, China
School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China

Keywords:

entire function;  sharing value;  uniqueness

PACS:

O175.26

DOI:

10.3969/j.issn.1003-7985.2004.03.025

Abstract:

We study the uniqueness of entire functions and prove the following theorem: Let f(z) and g(z) be two nonconstant entire functions; n and k two positive integers with n>2k+4. If the zeros of both f(z) and g(z) are of multiplicity at least n, and f^(k)(z) and g^(k)(z) share 1 CM, then either f(z)=c₁e^cz, g(z)=c₂e^-cz, where c₁, c₂ and c are three constants satisfying (-1)^kc₁c₂c^2k= 1, or f(z)≡g(z).

References:

[1] Hayman W K. Meromorphic functions [M]. Oxford: Clarendon Press, 1964. 47; 57.
[2] Yang L. Value distribution theory [M]. Berlin: Spring-Verlag, 1993.
[3] Hayman W K. Picard values of meromorphic functions and their derivatives [J]. Ann Math, 1959, 70: 9-42.
[4] Clunie J. On a result of Hayman [J]. J London Math Soc, 1967, 42: 389-392.
[5] Fang M L, Hua X H. Entire functions that share one value [J]. Journal of Nanjing University Mathematical Biquarterly, 1996, 13(1): 44-48.(in Chinese)
[6] Yang C C, Hua X H. Uniqueness and value-sharing of meromorphic functions [J]. Ann Acad Sci Fenn Math, 1997, 22(2): 395-406.
[7] Chen H H. Yoshida functions and Picard values of integral functions and their derivatives [J]. Bull Austral Math Soc, 1996, 54: 373-381.
[8] Wang Y F. On Mues conjecture and Picard values [J]. Science in China, 1993, 36(1): 28-35.
[9] Wang Y F, Fang M L. Picard values and normal families of meromorphic functions with multiple zeroes [J]. Acta Math Sinica, New Series, 1998, 14(1): 17-26.
[10] Zhang M Z. On the uniqueness of entire functions concerning the multiplicity of zeros [J]. Journal of Nanjing University Mathematical Biquarterly, 2001, 18(2): 261-270.(in Chinese)
[11] Xu Y, Qiu H L. Entire functions sharing one value IM [J]. Indian J Pure and Appl Math, 2002, 31(7): 849-855.
[12] Yi H X, Yang C C. Unicity theory of meromorphic functions[M]. Beijing: Science Press, 1995. 41-42.(in Chinese)
[13] Frank G. Eine vermutung von Hayman uber nullstellen meromorpher funktion [J]. Math Z, 1976, 149: 29-36.

Memo

Memo:

Biography: Zhang Minzhu(1976—), female, master, zmzwl@hotmail.com.

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