On zeros and growth of solutions of complex difference equations

Abstract

Abstract Let f be an entire function of finite order, let n ≥ 1 $n\geq 1$ , m ≥ 1 $m\geq 1$ , L ( z , f ) ≢ 0 $L(z,f)\not \equiv 0$ be a linear difference polynomial of f with small meromorphic coefficients, and P d ( z , f ) ≢ 0 $P_{d}(z,f)\not \equiv 0$ be a difference polynomial in f of degree d ≤ n − 1 $d\leq n-1$ with small meromorphic coefficients. We consider the growth and zeros of f n ( z ) L m ( z , f ) + P d ( z , f ) $f^{n}(z)L^{m}(z,f)+P_{d}(z,f)$ . 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 α 1 z + p 2 e α 2 z $f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$ , where n ≥ 2 $n\geq 2$ , P d ( z , f ) ≢ 0 $P_{d}(z,f)\not \equiv 0$ is a difference polynomial in f of degree d ≤ n − 2 $d\leq n-2$ with small mromorphic coefficients, p i $p_{i}$ , α i $\alpha _{i}$ ( i = 1 , 2 $i=1,2$ ) are nonzero constants such that α 1 ≠ α 2 $\alpha _{1}\neq \alpha _{2}$ . Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.

Keywords

• Difference equations
• Growth
• Zeros
• Meromorphic solution