AbstractLet f be an entire function of finite order, let $n\geq 1$
n
≥
1
, $m\geq 1$
m
≥
1
, $L(z,f)\not \equiv 0$
L
(
z
,
f
)
≢
0
be a linear difference polynomial of f with small meromorphic coefficients, and $P_{d}(z,f)\not \equiv 0$
P
d
(
z
,
f
)
≢
0
be a difference polynomial in f of degree $d\leq n-1$
d
≤
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 $f^{n}(z)+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}$
f
n
(
z
)
+
P
d
(
z
,
f
)
=
p
1
e
α
1
z
+
p
2
e
α
2
z
, where $n\geq 2$
n
≥
2
, $P_{d}(z,f)\not \equiv 0$
P
d
(
z
,
f
)
≢
0
is a difference polynomial in f of degree $d\leq n-2$
d
≤
n
−
2
with small mromorphic coefficients, $p_{i}$
p
i
, $\alpha _{i}$
α
i
($i=1,2$
i
=
1
,
2
) are nonzero constants such that $\alpha _{1}\neq \alpha _{2}$
α
1
≠
α
2
. Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018.
“Borel” lines for meromorphic solutions of the difference equation $y \left( {x + 1} \right) = y \left( x \right) + 1 + \lambda / y \left( x \right)$
