Uniqueness of difference-differential polynomials of meromorphic functions
UDC 517.9 We investigate the problems of uniqueness of difference-differential polynomials of finite-order meromorphic functions sharing a small function ignoring multiplicity and obtain some results that extend the results of K. Liu, X. L. Liu, and T. B. Cao.
Gespeichert in:
| Datum: | 2019 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2019
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1484 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507269183045632 |
|---|---|
| author | Dyavanal, R. S. Mathai, M. M. Дьяванал, Р. С. Матаі, М. М. |
| author_facet | Dyavanal, R. S. Mathai, M. M. Дьяванал, Р. С. Матаі, М. М. |
| author_sort | Dyavanal, R. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:57:08Z |
| description | UDC 517.9
We investigate the problems of uniqueness of difference-differential polynomials of finite-order meromorphic functions
sharing a small function ignoring multiplicity and obtain some results that extend the results of K. Liu, X. L. Liu, and
T. B. Cao. |
| first_indexed | 2026-03-24T02:06:37Z |
| format | Article |
| fulltext |
UDC 517.9
R. S. Dyavanal, M. M. Mathai (Karnatak Univ., India)
UNIQUENESS OF DIFFERENCE-DIFFERENTIAL POLYNOMIALS
OF MEROMORPHIC FUNCTIONS*
ПРО ЄДИНIСТЬ РIЗНИЦЕВО-ДИФЕРЕНЦIАЛЬНИХ ПОЛIНОМIВ
МЕРОМОРФНИХ ФУНКЦIЙ
We investigate the problems of uniqueness of difference-differential polynomials of finite-order meromorphic functions
sharing a small function ignoring multiplicity and obtain some results that extend the results of K. Liu, X. L. Liu, and
T. B. Cao.
Вивчаються проблеми єдиностi рiзницево-диференцiальних полiномiв мероморфних функцiй скiнченного порядку,
що подiляють малу функцiю (нехтуючи кратнiстю). Отримано деякi результати, що узагальнюють результати K. Liu,
X. L. Liu i T. B. Cao.
1. Introduction and results. In this paper, a meromorphic function always means it is meromor-
phic in the complex plane \BbbC . We assume that the reader is familiar with standard notations of the
Nevanlinna theory of entire and meromorphic functions as explained in [5, 6, 14].
Let f(z) and \alpha (z) be two meromorphic functions. We say that \alpha (z) is a small function with
respect to f(z) if T (r, \alpha (z)) = S(r, f), where S(r, f) is used to denote any quantity satisfying
S(r, f) = o
\bigl(
T (r, f)
\bigr)
as r \rightarrow \infty , outside an exceptional set E of finite logarithmic measure, i.e.,
\mathrm{l}\mathrm{i}\mathrm{m}r\rightarrow \infty
\int
(1,r]\cap E
dt
t
< \infty .
Let f(z) and g(z) be two non-constant meromorphic functions. If, for a \in \BbbC = \BbbC \cup \{ \infty \} , the
quantities f(z) - a and g(z) - a have the same set of zeros with the same multiplicities, then we
say that f(z) and g(z) share the value a CM (counting multiplicities). At the same time, if we
do not consider the multiplicities, then f(z) and g(z) are said to share the value a IM (ignoring
multiplicities). Let f(z) and g(z) share the value 1 IM and let z0 be a 1-point of f(z) of order p
and a 1-point of g(z) of order q. We denote the counting function of the 1-points of both f(z) and
g(z) with p > q by NL
\biggl(
r,
1
f - 1
\biggr)
. In the same way, we can define NL
\biggl(
r,
1
g - 1
\biggr)
.
Let f(z) be a non-constant meromorphic function. Let a be a finite complex number, and let k
be a positive integer. By Nk)
\biggl(
r,
1
f - a
\biggr) \biggl(
or Nk)
\biggl(
r,
1
f - a
\biggr) \biggr)
, we denote the counting function
of the roots of f(z) - a with multiplicity \leq k(IM) and by N(k
\biggl(
r,
1
f - a
\biggr) \biggl(
or N (k
\biggl(
r,
1
f - a
\biggr) \biggr)
,
we denote the counting function of the roots of f(z) - a with multiplicity \geq k (IM). We set
Nk
\biggl(
r,
1
f - a
\biggr)
= N
\biggl(
r,
1
f - a
\biggr)
+N (2
\biggl(
r,
1
f - a
\biggr)
+ . . .+N (k
\biggl(
r,
1
f - a
\biggr)
.
Further, we define the order \rho (f) of a meromorphic function f(z) by
* First author is supported by Ref. No. F. 510/3/DRS-III/2016(SAP-I) and the second author is supported by Ref.
No. KU/Sch/UGC-UPE/2014-15/894.
c\bigcirc R. S. DYAVANAL, M. M. MATHAI, 2019
906 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
UNIQUENESS OF DIFFERENCE-DIFFERENTIAL POLYNOMIALS OF MEROMORPHIC FUNCTIONS 907
\rho (f) = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow \infty
\mathrm{l}\mathrm{o}\mathrm{g} T (r, f)
\mathrm{l}\mathrm{o}\mathrm{g} r
,
and the hyper order \rho 2(f) of a meromorphic function f(z) by
\rho 2(f) = \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow \infty
\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g} T (r, f)
\mathrm{l}\mathrm{o}\mathrm{g} r
.
Let m be a non-negative integer, a0(\not = 0), a1, . . . , am - 1, am(\not = 0) be complex constants. Define
P (w) = amwm + am - 1w
m - 1 + . . .+ a1w + a0. (1.1)
In 2010, X. G. Qi, L. Z. Yang and K. Liu [12] considered the problems of uniqueness regarding
the difference polynomials of entire functions and obtained the following result.
Theorem A. Let f(z) and g(z) be transcendental entire functions of finite order and c be a
non-zero complex constant. If n \geq 6, f(z)nf(z + c) and g(z)ng(z + c) share 1 CM , then fg = t1
or f = t2g for some constants t1 and t2 that satisfy tn+1
1 = 1 and tn+1
2 = 1.
In 2011, X. M. Li, W. L. Li, H. X. Yi, Z. T. Wen [7] have improved the above result and obtained
the following result.
Theorem B. Let f(z) and g(z) be transcendental entire functions of finite order and \alpha (z) be
a meromorphic function such that \rho (\alpha ) < \rho (f), let c be a non-zero complex constant and let n \geq 7
be an integer. If f(z)n(f(z) - 1)f(z+ c) - \alpha (z) and g(z)n(g(z) - 1)g(z+ c) - \alpha (z) share 0 CM ,
then f(z) \equiv g(z).
Next, K. Liu, X. L. Liu, T. B. Cao [8 – 10] proved the following results.
Theorem C. Let f(z) and g(z) be transcendental meromorphic functions of finite order. Sup-
pose that c is a non-zero constant and n \in \BbbN . If n \geq 26, f(z)nf(z + c) and g(z)ng(z + c) share 1
IM , then f = tg or fg = t, where tn+1 = 1.
Theorem D. Let f(z) and g(z) be transcendental entire functions of finite order, n \geq 5k + 12.
If [f(z)nf(z + c)](k) and
\bigl[
g(z)ng(z + c)
\bigr] (k)
share the value 1 IM , then either f(z) = c1e
Cz,
g(z) = c2e
- Cz, where c1, c2 and C are constants satisfying ( - 1)k (c1c2)
n+1 \bigl[ (n+ 1)C
\bigr] 2k
= 1 or
f = tg, where tn+1 = 1.
Theorem E. Let f(z) and g(z) be transcendental entire functions of \rho 2(f) > 1, n \geq 5k +
+ 4m + 12. If
\bigl[
fn(fm - 1)f(z + c)
\bigr] (k)
and
\bigl[
gn(gm - 1)g(z + c)
\bigr] (k)
share the value 1 IM , then
f = tg, where tn+1 = tm = 1.
In this paper, we shall extend these results to meromorphic functions and obtain the following
two theorems.
Theorem 1.1. Let f(z) and g(z) be two non-constant finite order meromorphic functions. Sup-
pose that a(z)(\not \equiv 0,\infty ) is a small function with respect to f(z), which has no common zeros or
poles with f(z) and g(z). Let k(> 0) and m(> 0) be two integers satisfying n > 4m+ 13k + 19,
P (w) be as defined in (1.1) and c be a non-zero complex constant such that f(z) and g(z) are not
periodic functions of period c, poles of f(z) are not zeros of f(z + c) and poles of g(z) are not
zeros of g(z + c). If
\bigl[
fnP (f)f(z + c)
\bigr] (k)
and
\bigl[
gnP (g)g(z + c)
\bigr] (k)
share a(z) IM , f(z) and g(z)
share \infty IM , then one of the following two cases holds:
(1) f \equiv tg, for a constant t such that td = 1, where d = GCD(n +m + 1, . . . , n +m + 1 -
- i, . . . , n+ 1), am - i \not = 0 for some i = 0, 1, 2, . . . ,m;
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
908 R. S. DYAVANAL, M. M. MATHAI
(2) f(z) and g(z) satisfy the algebraic difference equation R(f, g) \equiv 0, where R(w1, w2) =
= wn
1 (amwm
1 + am - 1w
m - 1
1 + . . .+ a0)w1(z + c) - wn
2 (amwm
2 + am - 1w
m - 1
2 + . . .+ a0)w2(z + c).
Theorem 1.2. Let f(z) and g(z) be two non-constant finite order meromorphic functions. Sup-
pose that a(z)(\not \equiv 0,\infty ) is a small function with respect to f(z), which has no common zeros or
poles with f(z) and g(z). Let k(> 0) be integer satisfying n > 13k + 19, P (w) = a0, where
a0 \not = 0 is a complex constant and c be a non-zero complex constant such that f(z) and g(z) are
not periodic functions of period c, poles of f(z) are not zeros of f(z + c) and poles of g(z) are not
zeros of g(z + c). If
\bigl[
fnP (f)f(z + c)
\bigr] (k)
and
\bigl[
gnP (g)g(z + c)
\bigr] (k)
share a(z) IM , f(z) and g(z)
share \infty IM , then one of the following two cases holds:
(1) f(z) \equiv tg(z) for a constant t such that tn+1 = 1;
(2) a20
\bigl[
fnf(z + c)
\bigr] (k)\bigl[
gng(z + c)
\bigr] (k)
= a2(z).
2. Some lemmas. We need the following lemmas to prove our results.
Lemma 2.1 [2]. Let f(z) be a meromorphic function of finite order \rho and let c be a fixed
non-zero complex constant. Then, for each \epsilon > 0, we have
m
\biggl(
r,
f(z + c)
f(z)
\biggr)
+m
\biggl(
r,
f(z)
f(z + c)
\biggr)
= O(r\rho - 1+\epsilon ).
Lemma 2.2 [3]. Let f(z) be a meromorphic function of finite order \rho and let c be a fixed
non-zero complex constant. Then, for each \epsilon > 0, we have
T
\bigl(
r, f(z + c)
\bigr)
= T (r, f) +O(r\rho - 1+\epsilon ).
It is evident that S
\bigl(
r, f(z + c)
\bigr)
= S(r, f).
Lemma 2.3 [11]. Let f(z) be a meromorphic function of finite order \rho and let c be a fixed
non-zero complex constant. Then
(i) N
\biggl(
r,
1
f(z + c)
\biggr)
\leq N
\biggl(
r,
1
f
\biggr)
+ S(r, f),
(ii) N
\bigl(
r, f(z + c)
\bigr)
\leq N(r, f) + S(r, f),
(iii) N
\biggl(
r,
1
f(z + c)
\biggr)
\leq N
\biggl(
r,
1
f
\biggr)
+ S(r, f),
(iv) N
\bigl(
r, f(z + c)
\bigr)
\leq N(r, f) + S(r, f),
outside an exceptional set with finite logarithmic measure.
Lemma 2.4 [15]. Let f(z) be a non-constant meromorphic function and p, k be two positive
integers. Then
Np
\biggl(
r,
1
f (k)
\biggr)
\leq T
\Bigl(
r, f (k)
\Bigr)
- T (r, f) +Np+k
\biggl(
r,
1
f
\biggr)
+ S(r, f),
Np
\biggl(
r,
1
f (k)
\biggr)
\leq kN(r, f) +Np+k
\biggl(
r,
1
f
\biggr)
+ S(r, f).
Lemma 2.5 ([13], Lemma 3). Let f(z) and g(z) be two non-constant meromorphic functions.
If f(z) and g(z) share 1 CM , then one of the following three cases holds:
(1) T (r, f) \leq N2
\biggl(
r,
1
f
\biggr)
+N2
\biggl(
r,
1
g
\biggr)
+N2 (r, f) +N2 (r, g) + S(r, f) + S(r, g);
the same inequality holds for T (r, g);
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
UNIQUENESS OF DIFFERENCE-DIFFERENTIAL POLYNOMIALS OF MEROMORPHIC FUNCTIONS 909
(2) fg = 1;
(3) f \equiv g.
Lemma 2.6 [15]. Let f1(z) and f2(z) be two non-constant meromorphic functions. If c1f1 +
+ c2f2 = c3, where c1, c2 and c3 are non-zero constants, then
T (r, f1) \leq N(r, f1) +N
\biggl(
r,
1
f1
\biggr)
+N
\biggl(
r,
1
f2
\biggr)
+ S(r, f1).
Define
H =
\Biggl(
F
\prime \prime
F \prime - 2F
\prime
F - 1
\Biggr)
-
\Biggl(
G
\prime \prime
G\prime - 2G
\prime
G - 1
\Biggr)
, (2.1)
V =
\Biggl(
F
\prime
F - 1
- F
\prime
F
\Biggr)
-
\Biggl(
G
\prime
G - 1
- G
\prime
G
\Biggr)
, (2.2)
where F =
\bigl[
fnP (f)f(z + c)
\bigr] (k)
a(z)
and G =
\bigl[
gnP (g)g(z + c)
\bigr] (k)
a(z)
, both f(z) and g(z) are mero-
morphic functions of finite order, c is a non-zero complex constant such that f(z) and g(z) are not
periodic functions of period c, a(z)(\not \equiv 0,\infty ) be a small function with respect to both f(z) and g(z),
which has no common zeros or poles with f(z) and g(z).
Using the similar method as in Lemma 2.14 of Banerjee [1], we obtain the following lemma.
Lemma 2.7. Let F, G and H be defined as in (2.1). If F and G share 1 IM and \infty IM , and
H \not \equiv 0, then F \not \equiv G and
T (r, F ) \leq N2
\biggl(
r,
1
F
\biggr)
+N2
\biggl(
r,
1
G
\biggr)
+ 2N
\biggl(
r,
1
F
\biggr)
+N
\biggl(
r,
1
G
\biggr)
+ 7N(r, F ) + S(r, F ) + S(r,G),
the same inequality holds for T (r,G).
Lemma 2.8 [16]. Let F, G and V be defined as in (2.2). If F and G share \infty IM and V \equiv 0,
then F \equiv G.
Lemma 2.9 [16]. If F and G share 1 IM , then
NL
\biggl(
r,
1
F - 1
\biggr)
\leq N
\biggl(
r,
1
F
\biggr)
+N(r, F ) + S(r, F ) + S(r,G).
Lemma 2.10. Let f(z), g(z) be two non-constant finite order meromorphic functions such that
poles of f(z) are not zeros of f(z + c) and poles of g(z) are not zeros of g(z + c), F, G and V be
defined as in (2.2), P (w) be defined as in (1.1) and n(> 3), k(> 0), m(\geq 0) be three integers. Let
c be a non-zero complex constant such that f(z) and g(z) are not periodic functions of period c. If
V \not \equiv 0, F and G share 1 and \infty IM , then
(n+m+ k - 5)N(r, f) \leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+ S(r, f) + S(r, g)
and
(n+m+ k - 5)N(r, g) \leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+ S(r, f) + S(r, g).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
910 R. S. DYAVANAL, M. M. MATHAI
Proof. Let z0 be a pole of f(z) and g(z) with multiplicities p and q, respectively. By using
hypotheses V \not \equiv 0, F and G share \infty IM, pole of f(z) is not a zero of f(z + c) and a pole of g(z)
is not a zero of g(z + c), we get z0 is pole of F with multiplicity (n+m)p+ k and pole of G with
multiplicity (n+m)q + k.
Thus z0 is zero of
F
\prime
F - 1
- F
\prime
F
with multiplicity (n+m)p+ k - 1 \geq n+m+ k - 1 and also
z0 is zero of
G
\prime
G - 1
- G
\prime
G
with multiplicity (n+m)q+ k - 1 \geq n+m+ k - 1, hence z0 is zero of
V with multiplicity at least n+m+ k - 1. Thus
(n+m+ k - 1)N(r, f) \leq N
\biggl(
r,
1
V
\biggr)
(2.3)
and
(n+m+ k - 1)N(r, g) \leq N
\biggl(
r,
1
V
\biggr)
. (2.4)
By the lemma of the logarithmic derivative, we have
m(r, V ) = S(r, f) + S(r, g).
Now consider
N
\biggl(
r,
1
V
\biggr)
\leq T (r, V ) \leq m(r, V ) +N(r, V ) \leq N(r, V ) + S(r, f) + S(r, g). (2.5)
Since F (z) and G(z) share the value 1 IM, zeros of F (z) - 1 and zeros of G(z) - 1 of different
multiplicities contribute to poles of V and also since F (z) and G(z) share the value \infty IM, the
poles of F (z) and G(z) of different multiplicities contributes to zeros of V. Thus from (2.2) and
(2.5), we deduce
N
\biggl(
r,
1
V
\biggr)
\leq N
\biggl(
r,
1
F
\biggr)
+
+N
\biggl(
r,
1
G
\biggr)
+NL
\biggl(
r,
1
F - 1
\biggr)
+NL
\biggl(
r,
1
G - 1
\biggr)
+ S(r, f) + S(r, g). (2.6)
Since F and G share 1 IM, by Lemma 2.9 and (2.6), we get
N
\biggl(
r,
1
V
\biggr)
\leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+N(r, F ) +N(r,G) + S(r, f) + S(r, g). (2.7)
By Lemma 2.3, we obtain
N(r, F ) = N
\Biggl(
r,
\bigl[
fnP (f)f(z + c)
\bigr] (k)
a(z)
\Biggr)
\leq
\leq N(r, f) +N(r, f(z + c)) + S(r, f) \leq 2N(r, f) + S(r, f). (2.8)
Similarly,
N(r,G) \leq 2N(r, g) + S(r, g). (2.9)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
UNIQUENESS OF DIFFERENCE-DIFFERENTIAL POLYNOMIALS OF MEROMORPHIC FUNCTIONS 911
From (2.7) – (2.9) and using that f(z) and g(z) share \infty IM, we have
N
\biggl(
r,
1
V
\biggr)
\leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+ 2N(r, f) + 2N(r, g) + S(r, f) + S(r, g) \leq
\leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+ 4N(r, f) + S(r, f) + S(r, g). (2.10)
It follows from (2.3) and (2.10) that
(n+m+ k - 1)N(r, f) \leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+ 4N(r, f) + S(r, f) + S(r, g),
i.e.,
(n+m+ k - 5)N(r, f) \leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+ S(r, f) + S(r, g).
Similarly,
(n+m+ k - 5)N(r, g) \leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+ S(r, f) + S(r, g).
Lemma 2.11 [4]. Let f(z) be a non-constant finite order meromorphic function. Let P (f) be
as defined in (1.1) and c be a non-zero complex constant such that f(z) is not periodic function of
period c. Then
(n+m - 1)T (r, f) + S(r, f) \leq T (r, fnP (f)f(z + c)) \leq (n+m+ 1)T (r, f) + S(r, f).
Lemma 2.12 [4]. Let f(z) be a transcendental finite order meromorphic function. Let k(> 0)
be integer satisfying n > k + 5, c be a non-zero complex constant such that f(z) is not periodic
function of period c and let P (w) be as defined in (1.1). Suppose that a(z)(\not \equiv 0,\infty ) is a small
function with respect to f(z). Then (fnP (f)f(z + c))(k) - a(z) has infinitely many zeros.
Lemma 2.13 [4]. Let f(z) and g(z) be two non-constant finite order meromorphic functions.
Let P (w) be as defined in (1.1). Let k(> 0), m(\geq 0) be integers satisfying n > 2k +m + 5 and
c be a non-zero complex constant such that f(z) and g(z) are not periodic functions of period c. If\bigl[
fnP (f)f(z + c)
\bigr] (k) \equiv \bigl[ gnP (g)g(z + c)
\bigr] (k)
, then fnP (f)f(z + c) \equiv gnP (g)g(z + c).
Lemma 2.14 [4]. Let f(z) and g(z) be two non-constant finite order meromorphic functions.
Let c be a non-zero complex constant such that f(z) and g(z) are not periodic functions of period
c and k(> 0) be integer satisfying n > k + 5. Let P (w) be as defined in (1.1). Suppose that
a(z)(\not \equiv 0,\infty ) is a small function with respect to f(z) with finitely many zeros and poles. If\bigl(
fnP (f)f(z + c)
\bigr) (k)\bigl(
gnP (g)g(z + c)
\bigr) (k)
= a2(z), f(z) and g(z) share \infty IM , then P (w) re-
duces to a non-zero monomial, namely, P (w) = aiw
i \not \equiv 0 for some i \in \{ 0, 1, . . . ,m\} .
3. Proof of the theorems. 3.1. Proof of Theorem 1.1. Let F, G, H and V be as defined
in (2.1) and (2.2). If F1 = fnP (f)f(z + c) and G1 = gnP (g)g(z + c), then F and G share 1 and
\infty IM. Suppose that H \not \equiv 0. Then according to Lemmas 2.7 and 2.8, F \not \equiv G and V \not \equiv 0 and it
follows that
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
912 R. S. DYAVANAL, M. M. MATHAI
T (r, F ) \leq N2
\biggl(
r,
1
F
\biggr)
+N2
\biggl(
r,
1
G
\biggr)
+ 2N
\biggl(
r,
1
F
\biggr)
+N
\biggl(
r,
1
G
\biggr)
+ 7N(r, F )+
+S(r, F ) + S(r,G). (3.1)
By Lemma 2.4 with p = 2, Lemma 2.3 and (3.1), we obtain
T (r, F1) \leq N2
\biggl(
r,
1
G
\biggr)
+ 2N
\biggl(
r,
1
F
\biggr)
+N
\biggl(
r,
1
G
\biggr)
+Nk+2
\biggl(
r,
1
F1
\biggr)
+ 7N(r, F )+
+S(r, F ) + S(r,G) \leq
\leq Nk+2
\biggl(
r,
1
G1
\biggr)
+ kN(r,G1) + 2Nk+1
\biggl(
r,
1
F1
\biggr)
+ 2kN(r, F1) +Nk+1
\biggl(
r,
1
G1
\biggr)
+
+kN(r,G1) +Nk+2
\biggl(
r,
1
F1
\biggr)
+ 7N(r, F ) + S(r, F ) + S(r,G) \leq
\leq (k + 2)N
\biggl(
r,
1
g
\biggr)
+N
\biggl(
r,
1
P (g)
\biggr)
+N
\biggl(
r,
1
g(z + c)
\biggr)
+ 2kN(r, g)+
+2(k + 1)N
\biggl(
r,
1
f
\biggr)
+ 2N
\biggl(
r,
1
P (f)
\biggr)
+ 2N
\biggl(
r,
1
f(z + c)
\biggr)
+ 4kN(r, f)+
+(k + 1)N
\biggl(
r,
1
g
\biggr)
+N
\biggl(
r,
1
P (g)
\biggr)
+N
\biggl(
r,
1
g(z + c)
\biggr)
+ 2kN(r, g)+
+(k + 2)N
\biggl(
r,
1
f
\biggr)
+N
\biggl(
r,
1
P (f)
\biggr)
+N
\biggl(
r,
1
f(z + c)
\biggr)
+
+14N(r, f) + S(r, f) + S(r, g),
i.e.,
T (r, F1) \leq (3k + 4)N
\biggl(
r,
1
f
\biggr)
+ (2k + 3)N
\biggl(
r,
1
g
\biggr)
+ 3N
\biggl(
r,
1
P (f)
\biggr)
+ 2N
\biggl(
r,
1
P (g)
\biggr)
+
+3N
\biggl(
r,
1
f
\biggr)
+ 2N
\biggl(
r,
1
g
\biggr)
+ (8k + 14)N(r, f) + S(r, f) + S(r, g).
By Lemma 2.11, the above inequality reduces
(n+m - 1)T (r, f) \leq (3k + 3m+ 7)T (r, f) + (2k + 2m+ 5)T (r, g) + (8k + 14)N(r, f)+
+S(r, f) + S(r, g). (3.2)
Similarly,
(n+m - 1)T (r, g) \leq (3k + 3m+ 7)T (r, g) + (2k + 2m+ 5)T (r, f) + (8k + 14)N(r, f)+
+S(r, f) + S(r, g). (3.3)
From (3.2) and (3.3), we get
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
UNIQUENESS OF DIFFERENCE-DIFFERENTIAL POLYNOMIALS OF MEROMORPHIC FUNCTIONS 913
(n+m - 1)(T (r, f) + T (r, g)) \leq (5k + 5m+ 12)(T (r, f) + T (r, g)) + 2(8k + 14)N(r, f)+
+S(r, f) + S(r, g),
i.e.,
(n - 4m - 5k - 13)
\bigl(
T (r, f) + T (r, g)
\bigr)
\leq 2(8k + 14)N(r, f) + S(r, f) + S(r, g). (3.4)
Since V \not \equiv 0, F and G share 1 and \infty IM, by Lemma 2.10, we have
(n+m+ k - 5)N(r, f) \leq 2N
\biggl(
r,
1
F
\biggr)
+ 2N
\biggl(
r,
1
G
\biggr)
+ S(r, f) + S(r, g). (3.5)
By Lemma 2.4 with p = 1, (3.5) reduces
(n+m+ k - 5)N(r, f) \leq 2(k + 1)N
\biggl(
r,
1
f
\biggr)
+ 2N
\biggl(
r,
1
P (f)
\biggr)
+ 2N
\biggl(
r,
1
f(z + c)
\biggr)
+
+2kN(r, f) + 2kN(r, f(z + c)) + 2(k + 1)N
\biggl(
r,
1
g
\biggr)
+
+2N
\biggl(
r,
1
P (g)
\biggr)
+ 2N
\biggl(
r,
1
g(z + c)
\biggr)
+ 2kN(r, g)+
+2kN(r, g(z + c)) + S(r, f) + S(r, g) \leq
\leq 2(k +m+ 2)T (r, f) + 2(k +m+ 2)T (r, g) + 8kN(r, f) + S(r, f) + S(r, g),
i.e.,
(n+m - 7k - 5)N(r, f) \leq 2(k +m+ 2) [T (r, f) + T (r, g)] + S(r, f) + S(r, g). (3.6)
It follows from (3.4) and (3.6) that
[(n - 4m - 5k - 13)(n+m - 7k - 5) - 4(8k + 14)(k +m+ 2)][T (r, f) + T (r, g)] \leq
\leq S(r, f) + S(r, g),
which is a contradiction because n > 4m+ 13k + 19. Thus, H \equiv 0.
Similar to the proof of Lemma 2.5 applied to the functions F and G, we obtain the following cases:
(i) T (r, F ) \leq N2
\biggl(
r,
1
F
\biggr)
+N2
\biggl(
r,
1
G
\biggr)
+N2(r, F ) +N2(r,G) + S(r, F ) + S(r,G),
(ii) FG \equiv 1,
(iii) F \equiv G.
By the condition on n, the case (i) is impossible.
By Lemma 2.14, the case (ii) is impossible.
Hence, we get only the case (iii), i.e.,
\bigl[
fnP (f)f(z + c)
\bigr] (k) \equiv
\bigl[
gnP (g)g(z + c)
\bigr] (k)
, then, by
Lemma 2.13, we obtain fnP (f)f(z + c) \equiv gnP (g)g(z + c), i.e.,
fn
\bigl(
amfm + am - 1f
m - 1 + . . .+ a1f + a0
\bigr)
f(z + c) \equiv
\equiv gn
\bigl(
amgm + am - 1g
m - 1 + . . .+ a1g + a0
\bigr)
g(z + c). (3.7)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
914 R. S. DYAVANAL, M. M. MATHAI
Let h =
f
g
. If h is a constant then substituting f = gh and f(z+ c) = g(z+ c)h(z+ c) in (3.7),
we deduce amgn+m
\bigl(
hn+mh(z+c) - 1
\bigr)
g(z+c)+am - 1g
n+m - 1
\bigl(
hn+m - 1h(z+c) - 1
\bigr)
g(z+c)+ . . .
. . .+a1g
n+1
\bigl(
hn+1h(z+c) - 1
\bigr)
g(z+c)+a0g
n
\bigl(
hnh(z+c) - 1
\bigr)
g(z+c) \equiv 0, which implies hd = 1,
where d = GCD(n +m + 1, . . . , n +m + 1 - i, . . . , n + 1), am - i \not = 0 for i = 0, 1, . . . ,m. Thus
f(z) \equiv tg(z) for a constant t such that td = 1, where d = GCD(n+m+1, . . . , n+m+1 - i, . . .
. . . , n + 1), am - i \not = 0 for i = 0, 1, . . . ,m, which is the conclusion (1) of Theorem 1.1. If h
is not a constant then f(z) and g(z) satisfy the algebraic difference equation R(f, g) \equiv 0, where
R(w1, w2) = wn
1 (amwm
1 + am - 1w
m - 1
1 + . . . + a0)w1(z + c) - wn
2 (amwm
2 + am - 1w
m - 1
2 + . . .
. . .+ a0)w2(z + c), which is the conclusion (2) of Theorem 1.1.
3.2. Proof of Theorem 1.2. Substituting a1 = a2 = . . . = am = 0 in P (w) and proceeding as
in the proof of Theorem 1.1, we complete the proof of Theorem 1.2.
References
1. Banerjee A. Meromorphic functions sharing one value // Int. J. Math. Sci. – 2005. – 22. – P. 3587 – 3598.
2. Bergweiler W., Langley J. K. Zeros of differences of meromorphic functions // Math. Proc. Cambridge Phil. Soc. –
2007. – 142. – P. 133 – 147.
3. Chiang Y. M., Feng S. J. On the Nevanlinna characteristic of f(z + \eta ) and difference equations in the complex
plane // Ramanujan J. – 2008. – 16. – P. 105 – 129.
4. Dyavanal R. S., Mathai M. M. Uniqueness of difference-differential polynomials of meromorphic functions and its
applications // Indian J. Math. and Math. Sci. – 2016. – 12, № 1. – P. 11 – 30.
5. Hayman W. K. Meromorphic functions. – Oxford: Clarendon Press, 1964.
6. Laine I. Nevanlinna theory and complex differential equations. – Berlin: De Gruyter, 1993.
7. Li X. M., Li W. L., Yi H. X., Wen Z. T. Uniqueness theorems of entire functions whose difference polynomials share a
meromorphic function of a smaller order // Ann. Polon. Math. – 2011. – 102, № 2. – P. 111 – 127.
8. Liu K., Liu X. L., Cao T. B. Value distribution and uniqueness of difference polynomials // Appl. Math. J. Chinese
Univ. – 2011. – Article ID 234215. – 12 p.
9. Liu K., Liu X. L., Cao T. B. Some results on zeros and uniqueness of difference-differential polynomials // Appl.
Math. J. Chinese Univ. – 2012. – 27, № 1. – P. 94 – 104.
10. Liu K., Liu X. L., Cao T. B. Some results on zeros distributions and uniqueness of derivatives of difference
polynomials // arXiv:1107.0773 [math.CV] (2011).
11. Luo X., Lin W. C. Value sharing results for shifts of meromorphic functions // J. Math. Anal. and Appl. – 2011. –
377. – P. 441 – 449.
12. Qi X. G., Yang L. Z., Liu K. Uniqueness and periodicity of meromorphic functions concerning the difference operator //
Comput. Math. Appl. – 2010. – 60, № 6. – P. 1739 – 1746.
13. Yang C. C., Hua X. H. Uniqueness and value-sharing of meromorphic functions // Ann. Acad. Sci. Fenn. Math. –
1997. – 22. – P. 395 – 406.
14. Yang C. C., Yi H. X. Uniqueness theory of meromorphic functions. – Kluwer Acad. Publ., 2003.
15. Yi H. X. Uniqueness of meromorphic functions and a question of C. C. Yang // Complex Var. – 1990. – 14. –
P. 169 – 176.
16. Yi H. X. Meromorphic functions that share three sets // Kodai Math. J. – 1997. – 20. – P. 22 – 32.
Received 22.07.16,
after revision — 20.04.17
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
|
| id | umjimathkievua-article-1484 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T02:06:37Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c1/dca6563f6ce9460d316c13f8d6c634c1.pdf |
| spelling | umjimathkievua-article-14842019-12-05T08:57:08Z Uniqueness of difference-differential polynomials of meromorphic functions Про єдинiсть рiзницево-диференцiальних полiномiв мероморфних функцiй Dyavanal, R. S. Mathai, M. M. Дьяванал, Р. С. Матаі, М. М. UDC 517.9 We investigate the problems of uniqueness of difference-differential polynomials of finite-order meromorphic functions sharing a small function ignoring multiplicity and obtain some results that extend the results of K. Liu, X. L. Liu, and T. B. Cao. УДК 517.9 Вивчаються проблеми єдиностi рiзницево-диференцiальних полiномiв мероморфних функцiй скiнченного порядку, що подiляють малу функцiю (нехтуючи кратнiстю). Отримано деякi результати, що узагальнюють результати K. Liu, X. L. Liu i T. B. Cao. Institute of Mathematics, NAS of Ukraine 2019-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1484 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 7 (2019); 906-914 Український математичний журнал; Том 71 № 7 (2019); 906-914 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/1484/468 Copyright (c) 2019 Dyavanal R. S.; Mathai M. M. |
| spellingShingle | Dyavanal, R. S. Mathai, M. M. Дьяванал, Р. С. Матаі, М. М. Uniqueness of difference-differential polynomials of meromorphic functions |
| title | Uniqueness of difference-differential polynomials of meromorphic
functions |
| title_alt | Про єдинiсть рiзницево-диференцiальних полiномiв
мероморфних функцiй |
| title_full | Uniqueness of difference-differential polynomials of meromorphic
functions |
| title_fullStr | Uniqueness of difference-differential polynomials of meromorphic
functions |
| title_full_unstemmed | Uniqueness of difference-differential polynomials of meromorphic
functions |
| title_short | Uniqueness of difference-differential polynomials of meromorphic
functions |
| title_sort | uniqueness of difference-differential polynomials of meromorphic
functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1484 |
| work_keys_str_mv | AT dyavanalrs uniquenessofdifferencedifferentialpolynomialsofmeromorphicfunctions AT mathaimm uniquenessofdifferencedifferentialpolynomialsofmeromorphicfunctions AT dʹâvanalrs uniquenessofdifferencedifferentialpolynomialsofmeromorphicfunctions AT mataímm uniquenessofdifferencedifferentialpolynomialsofmeromorphicfunctions AT dyavanalrs proêdinistʹriznicevodiferencialʹnihpolinomivmeromorfnihfunkcij AT mathaimm proêdinistʹriznicevodiferencialʹnihpolinomivmeromorfnihfunkcij AT dʹâvanalrs proêdinistʹriznicevodiferencialʹnihpolinomivmeromorfnihfunkcij AT mataímm proêdinistʹriznicevodiferencialʹnihpolinomivmeromorfnihfunkcij |