Unicity of meromorphic functions concerning differences and small functions

Abstract

Abstract In this paper, we study the unicity of meromorphic functions concerning differences and small functions and mainly prove two results: 1. Let f f be a transcendental entire function of finite order with a Borel exceptional entire small function a ( z ) a\left(z) , and let η \eta be a constant such that Δ η 2 f ≢ 0 {\Delta }_{\eta }^{2}\hspace{0.25em}f\not\equiv 0 . If Δ η 2 f {\Delta }_{\eta }^{2}\hspace{0.25em}f and Δ η f {\Delta }_{\eta }\hspace{0.25em}f share Δ η a {\Delta }_{\eta }a CM, then a ( z ) a\left(z) is a constant a a and f ( z ) = a + B e A z f\left(z)=a+B{e}^{Az} , where A , B A,B are two nonzero constants; 2. Let f f be a transcendental meromorphic function with ρ 2 ( f ) < 1 {\rho }_{2}(f)\lt 1 , let a 1 {a}_{1} , a 2 {a}_{2} be two distinct small functions of f f , let L ( z , f ) L\left(z,f) be a linear difference polynomial, and let a 1 ≢ L ( z , a 2 ) {a}_{1}\not\equiv L\left(z,{a}_{2}) . If δ ( a 2 , f ) > 0 \delta \left({a}_{2},f)\gt 0 , and f f and L ( z , f ) L\left(z,f) share a 1 {a}_{1} and ∞ \infty CM, then L ( z , f ) − a 1 f − a 1 = c , \frac{L\left(z,f)-{a}_{1}}{f-{a}_{1}}=c, for some constant c ≠ 0 c\ne 0 . The results improve some results following C. X. Chen and R. R. Zhang [Uniqueness theorems related difference operators of entire functions, Chinese Ann. Math. Ser. A 42 (2021), no. 1, 11–22] and R. R. Zhang, C. X. Chen, and Z. B. Huang [Uniqueness on linear difference polynomials of meromorphic functions, AIMS Math. 6 (2021), no. 4, 3874–3888].