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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 $\sigma(d)<\sigma(f)$.
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참고문헌 (16)
참고문헌 신청
R. Bruck / 1996 / On entire functions which share one value CM with their first derivative / Results Math 30(1-2):21~24
W. Bergweiler / 2007 / Zeros of differences of meromorphic functions / Math. Proc. Cambridge Philos. Soc 142(1):133~147
C. X. Chen / 2014 / Entire functions and their high order differences / Taiwanese J. Math 18(3):711~729
Z. X. Chen / 2004 / On conjecture of R. Br¨uck, concerning the entire function sharing one value CM with its derivative / Taiwanese J. Math 8(2):235~244
Y. M. Chiang / 2008 / On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane / Ramanujan J 16(1):105~129
W. Bergweiler / 2007 / Zeros of differences of meromorphic functions / Math. Proc. Cambridge Philos. Soc 142(1):133~147
C. X. Chen / 2014 / Entire functions and their high order differences / Taiwanese J. Math 18(3):711~729
Z. X. Chen / 2004 / On conjecture of R. Br¨uck, concerning the entire function sharing one value CM with its derivative / Taiwanese J. Math 8(2):235~244
Y. M. Chiang / 2008 / On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane / Ramanujan J 16(1):105~129
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