Shared values and Borel exceptional values for high order difference operators
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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 $\sigma(d)<\sigma(f)$.
Abstract
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 $\sigma(d)<\sigma(f)$.
참고문헌 (16)
- R. Bruck, "On entire functions which share one value CM with their first derivative", Results Math, 30(1-2), pp.21~24 (1996)
- W. Bergweiler, "Zeros of differences of meromorphic functions", Math. Proc. Cambridge Philos. Soc, 142(1), pp.133~147 (2007)
- C. X. Chen, "Entire functions and their high order differences", Taiwanese J. Math, 18(3), pp.711~729 (2014)
- Z. X. Chen, "On conjecture of R. Br¨uck, concerning the entire function sharing one value CM with its derivative", Taiwanese J. Math, 8(2), pp.235~244 (2004)
- Y. M. Chiang, "On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane", Ramanujan J, 16(1), pp.105~129 (2008)
- Y. M. Chiang, "On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions", Trans. Amer. Math. Soc, 361(7), pp.3767~3791 (2009)
- W. K. Hayman, "Slowly growing integral and subharmonic functions", Comment. Math. Helv, 34, pp.75~84 (1960)
- W. K. Hayman, "Meromorphic Functions" (1964), Clarendon Press
- J. Heittokangas, "Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity", J. Math. Anal. Appl, 355(1), pp.352~363 (2009)
- G. Gundersen, "Correction to meromorphic functions that share four values", Trans. Amer. Math. Soc, 304(2), pp.847~850 (1987)
- G. Gundersen, "Finite order solutions of second order linear differential equations", Trans. Amer. Math. Soc, 305(1), pp.415~429 (1988)
- G. Gundersen, "Entire functions that share one values with one or two of their derivatives", J. Math. Anal. Appl, 233(1), pp.88~95 (1998)
- E. Mues, "Meromorphic functions sharing four valus", Complex Variables Theory Appl, 12(1-4), pp.167~179 (1989)
- C. C. Yang, "Uniqueness Theory of Meromorphic Functions" (2006), Science Press
- L. Yang, "Value Distribution Theory" (1993), Springer-Verlag & Science Press
- J. Zhang, "Entire functions sharing a small entire function with their difference operators", Bull. Iran. Math. Soc