| Summary: | In this paper, we study the unicity of meromorphic functions concerning differences and small functions and mainly prove two results: 1. Let ff 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 Δη2f≢0{\Delta }_{\eta }^{2}\hspace{0.25em}f\not\equiv 0. If Δη2f{\Delta }_{\eta }^{2}\hspace{0.25em}f and Δηf{\Delta }_{\eta }\hspace{0.25em}f share Δηa{\Delta }_{\eta }a CM, then a(z)a\left(z) is a constant aa and f(z)=a+BeAzf\left(z)=a+B{e}^{Az}, where A,BA,B are two nonzero constants; 2. Let ff be a transcendental meromorphic function with ρ2(f)<1{\rho }_{2}(f)\lt 1, let a1{a}_{1}, a2{a}_{2} be two distinct small functions of ff, let L(z,f)L\left(z,f) be a linear difference polynomial, and let a1≢L(z,a2){a}_{1}\not\equiv L\left(z,{a}_{2}). If δ(a2,f)>0\delta \left({a}_{2},f)\gt 0, and ff and L(z,f)L\left(z,f) share a1{a}_{1} and ∞\infty CM, then L(z,f)−a1f−a1=c,\frac{L\left(z,f)-{a}_{1}}{f-{a}_{1}}=c, for some constant c≠0c\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].
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