Entire functions that share two pairs of small functions

• Main Authors: Huang Xiaohuang, Deng Bingmao, Fang Mingliang
• Format: Article
• Language: English
• Published: De Gruyter 2021-05-01
• Series: Open Mathematics
• Subjects: unicity; entire functions; derivatives; small functions; 30d35; 39a32
• Online Access: https://doi.org/10.1515/math-2021-0011

In this paper, we study the unicity of entire functions and their derivatives and obtain the following result: let ff be a non-constant entire function, let a1{a}_{1}, a2{a}_{2}, b1{b}_{1}, and b2{b}_{2} be four small functions of ff such that a1≢b1{a}_{1}\not\equiv {b}_{1}, a2≢b2{a}_{2}\not\equiv {b}_{2}, and none of them is identically equal to ∞\infty . If ff and f(k){f}^{\left(k)} share (a1,a2)\left({a}_{1},{a}_{2}) CM and share (b1,b2)\left({b}_{1},{b}_{2}) IM, then (a2−b2)f−(a1−b1)f(k)≡a2b1−a1b2\left({a}_{2}-{b}_{2})f-\left({a}_{1}-{b}_{1}){f}^{\left(k)}\equiv {a}_{2}{b}_{1}-{a}_{1}{b}_{2}. This extends the result due to Li and Yang [Value sharing of an entire function and its derivatives, J. Math. Soc. Japan. 51 (1999), no. 7, 781–799].