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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 407351, 8 pages
http://dx.doi.org/10.1155/2012/407351
Research Article

Uniqueness Theorems on Difference Monomials of Entire Functions

1Shandong Transport Vocational College, Weifang, Shandong 261206, China
2Laiwu Vocational and Technical College, Laiwu, Shandong 271100, China
3Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Received 5 April 2012; Accepted 11 June 2012

Academic Editor: Matti Vuorinen

Copyright © 2012 Gang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this paper is to discuss the uniqueness of the difference monomials 𝑓 𝑛 𝑓 ( 𝑧 + 𝑐 ) . It assumed that 𝑓 and 𝑔 are transcendental entire functions with finite order and 𝐸 𝑘 ) ( 1 , 𝑓 𝑛 𝑓 ( 𝑧 + 𝑐 ) ) = 𝐸 𝑘 ) ( 1 , 𝑔 𝑛 𝑔 ( 𝑧 + 𝑐 ) ) , where 𝑐 is a nonzero complex constant and 𝑛 , 𝑘 are integers. It is proved that if one of the following holds (i) 𝑛 6 and 𝑘 = 3 , (ii) 𝑛 7 and 𝑘 = 2 , and (iii) 𝑛 1 0 and 𝑘 = 1 , then 𝑓 𝑔 = 𝑡 1 or 𝑓 = 𝑡 2 𝑔 for some constants 𝑡 2 and 𝑡 3 which satisfy 𝑡 2 𝑛 + 1 = 1 and 𝑡 3 𝑛 + 1 = 1 . It is an improvement of the result of Qi, Yang and Liu.