Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: meromorphic function, derivative, small function

MSC numbers: Primary 30D35