Bull. Korean Math. Soc. 2016; 53(1): 49-60
Printed January 31, 2016
https://doi.org/10.4134/BKMS.2016.53.1.49
Copyright © The Korean Mathematical Society.
Liangwen Liao and Jie Zhang
Nanjing University, China University of Mining and Technology
In this paper, we investigate the high order difference counterpart of Br\"uck's conjecture, and we prove one result that for a transcendental entire function $f$ of finite order, which has a Borel exceptional function $a$ whose order is less than one, if $\Delta ^nf$ and $f$ share one small function $d$ other than $a$ CM, then $f$ must be form of $f(z)=a+ce^{\beta z},$ where $c$ and $\beta$ are two nonzero constants such that $\frac{d-\Delta^na}{d-a}= {(e^{\beta}-1)}^n.$ This result extends Chen's result from the case of $\sigma(d)<1$ to the general case of $\sigma(d)<\sigma(f)$.
Keywords: uniqueness, entire function, difference equation, order
MSC numbers: Primary 30D35, 34M10
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