Author:
Majumder Sujoy,Das Pradip
Abstract
UDC 517.5
We discuss the problem of uniqueness of a meromorphic function
f
(
z
)
,
which shares
a
1
(
z
)
,
a
2
(
z
)
,
and
a
3
(
z
)
CM with its shift
f
(
z
+
c
)
, where
a
1
(
z
)
,
a
2
(
z
)
,
and
a
3
(
z
)
are three
c
-periodic distinct small functions of
f
(
z
)
and
c
∈
ℂ
∖
{
0
}
. The obtained result improves the recent result of Heittokangas et al. [Complex Var. and Elliptic Equat., <strong>56</strong>, No. 1–4, 81–92 (2011)] by dropping the assumption about the order of
f
(
z
)
. In addition, we introduce a way of characterizing elliptic functions in terms of meromorphic functions sharing values with two of their shifts. Moreover, we show by a number of illustrating examples that our results are, in certain senses, best possible.
Publisher
SIGMA (Symmetry, Integrability and Geometry: Methods and Application)
Reference15 articles.
1. Y. M. Chiang, S. J. Feng, On the Nevanlinna characteristic of $f(z +η)$ and difference equations in the complex plane, Ramanujian J., 16, № 1, 105–129 (2008).
2. G. G. Gundersen, K. Tohge, Unique range sets for polynomials or rational functions, Progress in Analysis, 235–246 (2003).
3. R. G. Halburd, R. J. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., 31, № 2, 463–478 (2006).
4. R. Halburd, R. Korhonen, K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc., 366, 4267–4298 (2014).
5. W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford (1964).