Affiliation:
1. Department of Basic Courses, Guangzhou Maritime University , Guangzhou 510725 , China
2. School of Mathematics and Statistics, Hubei University of Sciences and Technology , Xianning, 437100 , China
Abstract
Abstract
Let
f
f
be a transcendental entire function of finite order with a Picard exceptional value
β
∈
C
\beta \in {\mathbb{C}}
,
q
∈
C
\
{
0
,
1
}
q\in {\mathbb{C}}\setminus \left\{0,1\right\}
and
c
c
are complex constants. The authors prove that
D
q
,
c
f
(
z
)
−
a
f
(
z
)
−
a
=
a
a
−
β
,
\frac{{{\mathfrak{D}}}_{q,c}f\left(z)-a}{f\left(z)-a}=\frac{a}{a-\beta },
if
D
q
,
c
f
(
z
)
{{\mathfrak{D}}}_{q,c}f\left(z)
and
f
(
z
)
f\left(z)
share value
a
(
≠
β
)
a\left(\ne \beta )
CM, where
D
q
,
c
f
(
z
)
=
f
(
q
z
+
c
)
−
f
(
z
)
(
q
−
1
)
z
+
c
{{\mathfrak{D}}}_{q,c}f\left(z)=\frac{f\left(qz+c)-f\left(z)}{\left(q-1)z+c}
is the Hahn difference operator. This result generalizes the related results of Zongxuan Chen [On the difference counterpart of Brück’s conjecture, Acta Math. Sci. 34B (2014), 653–659].