Let f be an entire function of finite order, let \(n\geq 1\), \(m\geq 1\), \(L(z,f)\not \equiv 0\) be a linear difference polynomial of f with small meromorphic coefficients, and \(P_{d}(z,f)\not \equiv 0\) be a difference polynomial in f of degree \(d\leq n-1\) with small meromorphic coefficients. We consider the growth and zeros of \(f^{n}(z)L^{m}(z,f)+P_{d}(z,f)\). And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type \(f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}\), where \(n\geq 2\), \(P_{d}(z,f)\not \equiv 0\) is a difference polynomial in f of degree \(d\leq n-2\) with small mromorphic coefficients, \(p_{i}\), \(\alpha _{i}\) (\(i=1,2\)) are nonzero constants such that \(\alpha _{1}\neq \alpha _{2}\). Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

令f是一个有限阶的完整函数，令\（n \ geq 1 \），\（m \ geq 1 \），\（L（z，f）\ not \ equiv 0 \）是一个线性差分多项式亚纯系数小的f和\（P_ {d}（z，f）\ not \ equiv 0 \）是亚纯系数小的度f（\ d \ leq n-1 \）的差分多项式。我们考虑\（f ^ {n}（z）L ^ {m}（z，f）+ P_ {d}（z，f）\）的增长和零点。并给出了一些反例以证明I. Laine（J. Math。Anal。Appl。469：808-826，2019）证明的定理3.1是无效的。另外，我们研究类型为差分方程的亚纯解\（f ^ {n}（z）+ P_ {d}（z，f）= p_ {1} e ^ {\ alpha _ {1} z} + p_ {2} e ^ {\ alpha _ {2} Z} \） ，其中\（N \ GEQ 2 \） ，\（P_ {d}（Z，F）\不\当量0 \）是有区别的多项式在˚F度\（d \当量的n-2 \ ）的亚纯系数小（\（p_ {i} \），\（\ alpha _ {i} \）（\（i = 1,2 \））是非零常数，因此\（\ alpha _ {1} \ neq \ alpha _ {2} \）。我们的结果是对Laine 2019，Latreuch 2017，Liu和Mao 2018的改进和补充。