A result on an open problem of L\"{u}, Li and Yang
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In this paper we deal with the open problem posed by L\"{u}, Li and Yang \cite{11a}. In fact, we prove the following result: Let $f(z)$ be a transcendental meromorphic function of finite order having finitely many poles, $c_{1},c_{2},\ldots, c_{n}\in\mathbb{C}\setminus\{0\}$ and $k, n\in\mathbb{N}$. Suppose $f^{n}(z)$, $f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n})$ share $0$ CM and $f^{n}(z)-Q_{1}(z)$, $(f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n}))^{(k)}-Q_{2}(z)$ share $(0,1)$, where $Q_{1}(z)$ and $Q_{2}(z)$ are non-zero polynomials. If $n\geq k+1$, then $(f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n}))^{(k)}\equiv \frac{Q_{2}(z)}{Q_{1}(z)}f^{n}(z)$. Furthermore, if $Q_{1}(z)\equiv Q_{2}(z)$, then $f(z)=c\;e^{\frac{\lambda}{n}z}$, where $c, \lambda\in\mathbb{C}\setminus\{0\}$ such that $e^{\lambda (c_{1}+c_{2}+\cdots+c_{n})}=1$ and $\lambda^{k}=1$. Also we exhibit some examples to show that the conditions of our result are the best possible.
Abstract
In this paper we deal with the open problem posed by L\"{u}, Li and Yang \cite{11a}. In fact, we prove the following result: Let $f(z)$ be a transcendental meromorphic function of finite order having finitely many poles, $c_{1},c_{2},\ldots, c_{n}\in\mathbb{C}\setminus\{0\}$ and $k, n\in\mathbb{N}$. Suppose $f^{n}(z)$, $f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n})$ share $0$ CM and $f^{n}(z)-Q_{1}(z)$, $(f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n}))^{(k)}-Q_{2}(z)$ share $(0,1)$, where $Q_{1}(z)$ and $Q_{2}(z)$ are non-zero polynomials. If $n\geq k+1$, then $(f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n}))^{(k)}\equiv \frac{Q_{2}(z)}{Q_{1}(z)}f^{n}(z)$. Furthermore, if $Q_{1}(z)\equiv Q_{2}(z)$, then $f(z)=c\;e^{\frac{\lambda}{n}z}$, where $c, \lambda\in\mathbb{C}\setminus\{0\}$ such that $e^{\lambda (c_{1}+c_{2}+\cdots+c_{n})}=1$ and $\lambda^{k}=1$. Also we exhibit some examples to show that the conditions of our result are the best possible.
참고문헌 (18)
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