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In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let $f$ be a transcendental entire function, $n$ and $k$ be two positive integers. If $f^{n}-Q_{1}$ and $(f^{n})^{(k)}-Q_{2}$ share $0$ CM, and $n\geq k+1$, then $(f^{n})^{(k)}\equiv \frac{Q_{2}}{Q_{1}}f^{n}$. Furthermore, if $Q_{1}=Q_{2}$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_{1}$, $Q_{2}$ are polynomials with $Q_{1}Q_{2}\not\equiv 0$, and $c$, $\lambda$ are non-zero constants such that $\lambda^{k}=1$.This result shows that the Conjecture given by W. L\"u, Q. Li and C. Yang [{{On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281--1289.}}] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.
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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
J. Clunie / 1962 / On integral and meromorphic functions / J. London Math. Soc 37:17~22
G. G. Gundersen / 1980 / Meromorphic functions that share finite values with their derivative / J. Math. Anal. Appl 75(2):441~446
G. G. Gundersen / 1998 / Entire functions that share one value with one or two of their derivatives / J. Math. Anal. Appl 223(1):88~95
W. K. Hayman / 1964 / Meromorphic Functions / Clarendon Press
J. Clunie / 1962 / On integral and meromorphic functions / J. London Math. Soc 37:17~22
G. G. Gundersen / 1980 / Meromorphic functions that share finite values with their derivative / J. Math. Anal. Appl 75(2):441~446
G. G. Gundersen / 1998 / Entire functions that share one value with one or two of their derivatives / J. Math. Anal. Appl 223(1):88~95
W. K. Hayman / 1964 / Meromorphic Functions / Clarendon Press
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