A result on a conjecture of W. L\"{u}, Q. Li and C. Yang
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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.
Abstract
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.
참고문헌 (16)
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