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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.
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참고문헌 (14)
참고문헌 신청
Rainer Brück / 1996 / On Entire Functions which Share One Value CM with their First Derivative / Results in Mathematics 30(1-2):21~24
TINGBIN CAO / 2016 / ON THE BRÜCK CONJECTURE / Bulletin of the Australian Mathematical Society 93(2):248~259
Zong-Xuan Chen / 2004 / ON CONJECTURE OF R. BR\"{U}CK CONCERNING THE ENTIRE FUNCTION SHARING ONE VALUE CM WITH ITS DERIVATIVE / Taiwanese Journal of Mathematics 8(2):235~244
Yik-Man Chiang / 2008 / On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane / The Ramanujan Journal 16(1):105~129
Gary G Gundersen / 1998 / Entire Functions That Share One Value with One or Two of Their Derivatives / Journal of Mathematical Analysis and Applications 223(1):88~95
TINGBIN CAO / 2016 / ON THE BRÜCK CONJECTURE / Bulletin of the Australian Mathematical Society 93(2):248~259
Zong-Xuan Chen / 2004 / ON CONJECTURE OF R. BR\"{U}CK CONCERNING THE ENTIRE FUNCTION SHARING ONE VALUE CM WITH ITS DERIVATIVE / Taiwanese Journal of Mathematics 8(2):235~244
Yik-Man Chiang / 2008 / On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane / The Ramanujan Journal 16(1):105~129
Gary G Gundersen / 1998 / Entire Functions That Share One Value with One or Two of Their Derivatives / Journal of Mathematical Analysis and Applications 223(1):88~95
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