Affiliation:
1. Department of Mathematics, Hangzhou Dianzi University , Hangzhou 310012 , China
2. School of Financial Mathematics & Statistics, Guangdong University of Finance , Guangzhou 510521 , China
Abstract
Abstract
In this paper, we study the unicity of entire functions and their derivatives and obtain the following result: let
f
f
be a non-constant entire function, let
a
1
{a}_{1}
,
a
2
{a}_{2}
,
b
1
{b}_{1}
, and
b
2
{b}_{2}
be four small functions of
f
f
such that
a
1
≢
b
1
{a}_{1}\not\equiv {b}_{1}
,
a
2
≢
b
2
{a}_{2}\not\equiv {b}_{2}
, and none of them is identically equal to
∞
\infty
. If
f
f
and
f
(
k
)
{f}^{\left(k)}
share
(
a
1
,
a
2
)
\left({a}_{1},{a}_{2})
CM and share
(
b
1
,
b
2
)
\left({b}_{1},{b}_{2})
IM, then
(
a
2
−
b
2
)
f
−
(
a
1
−
b
1
)
f
(
k
)
≡
a
2
b
1
−
a
1
b
2
\left({a}_{2}-{b}_{2})f-\left({a}_{1}-{b}_{1}){f}^{\left(k)}\equiv {a}_{2}{b}_{1}-{a}_{1}{b}_{2}
. This extends the result due to Li and Yang [Value sharing of an entire function and its derivatives, J. Math. Soc. Japan. 51 (1999), no. 7, 781–799].
Reference16 articles.
1. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
2. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers Group, Dordrecht, 2003.
3. L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 1993.
4. G. Brosch, Eindeutigkeitssätze für meromorphe funktionen, Doctoral thesis, Technical University of Aachen, 1989.
5. T. P. Czubiak and G. G. Gundersen, Meromorphic functions that share pairs of values, Complex Variables Theory Appl. 34 (1997), no. 1–2, 35–46.