ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS

doi:10.1016/S0252-9602(17)30037-1

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (3): 786-798.doi: 10.1016/S0252-9602(17)30037-1

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ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS

Ning CUI, Zong-Xuan CHEN

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received:2016-02-17 Online:2017-06-25 Published:2017-06-25
Supported by:
This research was supported by the Natural Science Foundation of Guangdong Province in China (2014A030313422,2016A030310106,2016A030313745).

Abstract

Abstract:

In this article, we mainly devote to proving uniqueness results for entire functions sharing one small function CM with their shift and difference operator simultaneously. Let f(z) be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f(z), f(z+c), and Δ_cⁿf(z) share 0 CM, then f(z + c) ≡ Af(z), where A(≠0) is a complex constant. Moreover, let a(z), b(z)(? 0) ∈ S(f) be periodic entire functions with period c and if f(z) -a(z), f(z + c) -a(z), Δ_cⁿf(z) -b(z) share 0 CM, then f(z + c) ≡ f(z).

Key words: Entire function, shifts, difference operators, shared values

Ning CUI, Zong-Xuan CHEN. ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS[J].Acta mathematica scientia,Series B, 2017, 37(3): 786-798.

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ENTIRE FUNCTIONS SHARING ONE SMALL FUNCTION CM WITH THEIR SHIFTS AND DIFFERENCE OPERATORS

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