Value distribution of differential-difference polynomials of meromorphic functions
We obtain the results on the deficiencies of differential-difference polynomials. These results can be regarded as differential difference analogs of some classical theorems on differential polynomials. In particular, an exact estimate of the deficiency of some differential-difference polynomials is...
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| author | Zhang, R. R. Чжан, Р. Р. |
| author_facet | Zhang, R. R. Чжан, Р. Р. |
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| description | We obtain the results on the deficiencies of differential-difference polynomials. These results can be regarded as differential difference
analogs of some classical theorems on differential polynomials. In particular, an exact estimate of the deficiency of some differential-difference polynomials is presented. We also give examples showing that these results are best possible in a certain sense. |
| first_indexed | 2026-03-24T02:08:15Z |
| format | Article |
| fulltext |
UDC 517.9
R. R. Zhang (Guangdong Univ. Education, China)
VALUE DISTRIBUTION OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS
OF MEROMORPHIC FUNCTIONS*
РОЗПОДIЛ ЗНАЧЕНЬ ДЛЯ ДИФЕРЕНЦIАЛЬНО-РIЗНИЦЕВИХ ПОЛIНОМIВ
МЕРОМОРФНИХ ФУНКЦIЙ
We obtain the results on the deficiencies of differential-difference polynomials. These results can be regarded as differential-
difference analogs of some classical theorems on differential polynomials. In particular, an exact estimate of the deficiency
of some differential-difference polynomials is presented. We also give examples showing that these results are best possible
in a certain sense.
Отримано результати щодо дефектiв диференцiально-рiзницевих полiномiв. Цi результати можна розглядати як
диференцiально-рiзницевi аналоги деяких класичних теорем для диференцiальних полiномiв. Зокрема, наведено
точну оцiнку для дефектiв деяких диференцiально-рiзницевих полiномiв. Також наведено приклади, якi показують,
що цi результати є, в певному сенсi, найкращими.
1. Introduction and results. Let f(z) be a meromorphic function in the complex plane \BbbC . We
assume that the reader is familiar with the basic notions of Nevanlinna’s theory (see [7]). We use
\sigma (f) to denote the order of growth of f(z); \lambda (f) and \lambda (1/f) to denote, respectively, the exponents
of convergence of zero and pole sequences of f(z). The hyper order of f(z) is defined by
\sigma 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
.
For a \in \BbbC , we use \delta (a, f) to denote the Nevanlinna deficiency of a respect to f(z). Moreover, we
denote by S(r, f) any real function of growth o(T (r, f)) as r \rightarrow \infty outside of a possible exceptional
set of finite logarithmic measure. A meromorphic function \alpha (z) is said to be a small function of
f(z), if T (r, \alpha ) = S(r, f).
The value distribution of differential polynomials has been discussed extensively and deeply. For
example, Doeringer [4] investigated the differential polynomial
\Psi = f(z)nQ(f) + P (f),
where f(z) is a transcendental entire function and Q(f), P (f) are differential polynomials in f(z)
with small meromorphic coefficients such that Q(f) \not \equiv 0, P (f) \not \equiv 0. He proved that
\mathrm{l}\mathrm{i}\mathrm{m}r\rightarrow \infty
N(r, 1/\Psi )
T (r,\Psi )
> 0 holds for n \geq 2 + \gamma , where \gamma denotes the degree of P (f).
In the past decade, difference polynomials have been discussed extensively and many results
have been obtained (see, e.g., [2, 8 – 10, 16, 17]). Lately, authors began to investigate differential-
difference polynomials (see, e.g., [12, 14, 18]) and differential-difference equations (see, e.g., [3,
13]). The study of the value distribution of differential-difference polynomials plays an important
role in the further study of differential-difference equations.
* This work was supported by Training Plan Fund of Outstanding Young Teachers of Higher Learning Institutions
of Guangdong Province in China (Yq20145084602) and Natural Science Foundation of Guangdong Province in China
(2016A030313745).
c\bigcirc R. R. ZHANG, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4 471
472 R. R. ZHANG
A differential-difference polynomial is a polynomial in f(z), its shifts, its derivatives and deriva-
tives of its shifts, that is an expression of the form
P (z, f) =
\sum
\lambda \in J
a\lambda (z)
\tau \lambda \prod
j=1
\{ f (\alpha \lambda ,j)(z + \delta \lambda ,j)\} \beta \lambda ,j , (1.1)
where J is an index set, \delta \lambda ,j are complex constants, \alpha \lambda ,j and \beta \lambda ,j are nonnegative integers, and
the coefficients a\lambda (z)(\not \equiv 0) are small meromorphic functions of f(z). The maximal total degree of
P (z, f) is defined by
\mathrm{d}\mathrm{e}\mathrm{g}f P = \mathrm{m}\mathrm{a}\mathrm{x}
\lambda \in J
\tau \lambda \sum
j=1
\beta \lambda ,j .
Zheng and Xu [18] investigated differential-difference polynomials
Q1(z, f) = F (f)P (z, f)
and
Q2(z, f) = F (f) + P (z, f),
where f(z) is a transcendental meromorphic function satisfying \sigma 2(f) < 1 and N(r, f) = S(r, f),
P (z, f) is a differential-difference polynomial of f(z), and F (f) = (fv + av - 1(z)f
v - 1 + . . .
. . . + a0(z))
u is a polynomial of f(z). For a small meromorphic function \alpha (z)(\not \equiv 0), they proved
the following results:
(i) If uv > \mathrm{d}\mathrm{e}\mathrm{g}f P and u \not = 1, then \delta (\alpha ,Q1) \leq 1 -
(u - 1)(uv - \mathrm{d}\mathrm{e}\mathrm{g}f P )
u(uv + \mathrm{d}\mathrm{e}\mathrm{g}f P )
< 1.
(ii) If
(u - 1)uv
2u - 1
> \mathrm{d}\mathrm{e}\mathrm{g}f P and u \not = 1, then \delta (\alpha ,Q2) \leq
1
u
+
2u - 1
u2v
\mathrm{d}\mathrm{e}\mathrm{g}f P < 1.
In this paper, we prove a differential-difference counterpart of the result of [4] and obtain Theorem
1.1 below, which improves the results of [18].
Theorem 1.1. Let f(z) be a transcendental meromorphic function satisfying \sigma 2(f) < 1 and
N(r, f) = S(r, f). Let F (z, f) = (av(z)f
v + av - 1(z)f
v - 1 + . . .+ a0(z))
u, where u, v are positive
integers, and aj(z), j = 0, 1, . . . , v, are small meromorphic functions of f(z) with av(z) \not \equiv 0.
Suppose that P1(z, f) and P2(z, f) (P1(z, f)P2(z, f) \not \equiv 0) are differential-difference polynomials
with small meromorphic coefficients. If (u - 1)v > \mathrm{d}\mathrm{e}\mathrm{g}f P2, then
\psi (z) = F (z, f)P1(z, f) + P2(z, f)
satisfies \delta (0, \psi ) \leq
\mathrm{d}\mathrm{e}\mathrm{g}f P1 + v + \mathrm{d}\mathrm{e}\mathrm{g}f P2
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
< 1.
Remark 1.1. Comparing Theorem 1.1 with the results of [18], we see that the condition of
Theorem 1.1 is weaker and the conclusion is stronger. Examples 1 and 2 below show that both
\delta (0, \psi ) =
\mathrm{d}\mathrm{e}\mathrm{g}f P1 + v + \mathrm{d}\mathrm{e}\mathrm{g}f P2
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
and \delta (0, \psi ) <
\mathrm{d}\mathrm{e}\mathrm{g}f P1 + v + \mathrm{d}\mathrm{e}\mathrm{g}f P2
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
in Theorem 1.1 may hold,
and Example 3 below shows that Theorem 1.1 is false, if (u - 1)v = \mathrm{d}\mathrm{e}\mathrm{g}f P2. So the result of
Theorem 1.1 is best possible in this sense.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
VALUE DISTRIBUTION OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS . . . 473
Example 1.1. Let f(z) = ez, F (z, f) = (f(z) + 1)3, P1(z, f) \equiv 1 and P2(z, f) = 3f \prime (z +
+ \pi i) - 1. Then u = 3, v = 1,\mathrm{d}\mathrm{e}\mathrm{g}f P1 = 0, \mathrm{d}\mathrm{e}\mathrm{g}f P2 = 1 and \psi (z) = F (z, f)P1(z, f) +P2(z, f) =
= e2z(ez + 3) satisfies \delta (0, \psi ) =
2
3
=
\mathrm{d}\mathrm{e}\mathrm{g}f P1 + v + \mathrm{d}\mathrm{e}\mathrm{g}f P2
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
.
Example 1.2. Let f(z) = ez, F (z, f) = (f(z)2 + 1)2, P1(z, f) = f \prime (z) and P2(z, f) =
= f \prime \prime (z + \pi i). Then u = 2, v = 2,\mathrm{d}\mathrm{e}\mathrm{g}f P1 = 1,\mathrm{d}\mathrm{e}\mathrm{g}f P2 = 1 and \psi (z) = F (z, f)P1(z, f) +
+ P2(z, f) = e3z(e2z + 2) satisfies \delta (0, \psi ) =
3
5
<
\mathrm{d}\mathrm{e}\mathrm{g}f P1 + v + \mathrm{d}\mathrm{e}\mathrm{g}f P2
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
=
4
5
.
Example 1.3. Take f(z) = ez + 1, F (z, f) = (f(z) + 1)2, P1(z, f) \equiv 1 and P2(z, f) =
= 4f \prime (z+\pi i) - 4. Then (u - 1)v = (2 - 1)\times 1 = \mathrm{d}\mathrm{e}\mathrm{g}f P2 and \psi (z) = F (z, f)P1(z, f)+P2(z, f) =
= e2z satisfies \delta (0, \psi ) = 1.
In the case where f(z) has two Borel exceptional values, we get the following theorem.
Theorem 1.2. Let f(z) be a transcendental meromorphic function with \sigma (f) < \infty . Let
F (z, f) = (av(z)f
v + av - 1(z)f
v - 1 + . . . + a0(z))
u, where u, v are positive integers, and aj(z),
j = 0, 1, . . . , v, satisfy \sigma (aj) < \sigma (f) and av(z) \not \equiv 0. Suppose that P1(z, f)(\not \equiv 0) and P2(z, f)
are differential-difference polynomials, and the growth orders of their coefficients are less than
\sigma (f). Suppose further that a,\infty are Borel exceptional values of f(z) such that F (z, a)P1(z, a) +
+ P2(z, a) \not \equiv 0. If uv > \mathrm{d}\mathrm{e}\mathrm{g}f P2 or P2(z, f) \equiv 0, then
\psi (z) = F (z, f)P1(z, f) + P2(z, f)
satisfies \delta (0, \psi ) \leq
\mathrm{d}\mathrm{e}\mathrm{g}f P1
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
< 1.
Remark 1.2. (1) Theorem 1.2 dose not remain valid, if we replace the Borel exceptional values
“a,\infty ” with two finite Borel exceptional values. Indeed, take f(z) = \mathrm{t}\mathrm{a}\mathrm{n} z, F (z, f) = f(z)3,
P1(z, f) = zf
\Bigl(
z +
\pi
2
\Bigr) 3
and P2(z, f) = - 2f(z + \pi )f
\Bigl(
z +
\pi
2
\Bigr)
+ z +
1
z
- 2. We have \psi (z) =
= F (z, f)P1(z, f) + P2(z, f) =
1
z
. Obviously, uv = 3\times 1 > \mathrm{d}\mathrm{e}\mathrm{g}f P2 = 2, i and - i are two Borel
exceptional values of f(z), and F (z, i)P1(z, i)+P2(z, i) =
1
z
\not \equiv 0, F (z, - i)P1(z, - i)+P2(z, - i) =
=
1
z
\not \equiv 0. But \psi (z) =
1
z
satisfies \delta (0, \psi ) = 1.
(2) Theorem 1.2 is false, if uv = \mathrm{d}\mathrm{e}\mathrm{g}f P2. Indeed, take f(z) = ez
2
, F (z, f) = f(z)2, P1(z, f) \equiv
\equiv 1 and P2(z, f) = - e
- 2z - 1
2z
f(z+1)f \prime (z)+ez. We have \psi (z) = F (z, f)P1(z, f)+P2(z, f) = ez.
Obviously, uv = 2 \times 1 = \mathrm{d}\mathrm{e}\mathrm{g}f P2, and the Borel exceptional value 0 satisfies F (z, 0)P1(z, 0) +
+ P2(z, 0) = ez \not \equiv 0. But \psi (z) = ez satisfies \delta (0, \psi ) = 1.
At last, we give an exact estimate of the deficiency \delta (0, P ) of differential-difference polynomial
(1.1). In order to collect together its all monomials having the same degree, we introduce the notation
Jl =
\left\{ \lambda \in J
\bigm| \bigm| \bigm| \tau \lambda \sum
j=1
\beta \lambda ,j = l
\right\} , (1.2)
where l = 0, 1, . . . ,\mathrm{d}\mathrm{e}\mathrm{g}f P.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
474 R. R. ZHANG
Theorem 1.3. Let f(z) be a transcendental meromorphic function satisfying \sigma 2(f) < 1 and
N(r, f) +N(r, 1/f) = S(r, f). (1.3)
Suppose that P (z, f) is a differential-difference polynomial of the form (1.1). Denote
\mathrm{m}\mathrm{a}\mathrm{x}
\left\{ l\bigm| \bigm| \bigm| \sum
\lambda \in Jl
a\lambda (z)
\tau \lambda \prod
j=1
\{ f (\alpha \lambda ,j)(z + \delta \lambda ,j)\} \beta \lambda ,j \not \equiv 0, l \in \{ 0, 1, . . . ,\mathrm{d}\mathrm{e}\mathrm{g}f P\}
\right\} = m, (1.4)
\mathrm{m}\mathrm{i}\mathrm{n}
\left\{ l\bigm| \bigm| \bigm| \sum
\lambda \in Jl
a\lambda (z)
\tau \lambda \prod
j=1
\{ f (\alpha \lambda ,j)(z + \delta \lambda ,j)\} \beta \lambda ,j \not \equiv 0, l \in \{ 0, 1, . . . ,\mathrm{d}\mathrm{e}\mathrm{g}f P\}
\right\} = k, (1.5)
where Jl is defined by (1.2). If m > k or m = k \geq 1, then \delta (0, P ) =
k
m
.
The following example illustrates Theorem 1.3.
Example 1.4. Let f(z) = ez and P (z, f) = f(z)4+f \prime (z)f(z+\pi i)2+f \prime (z)2, then m = 4, k = 2
satisfy (1.4) and (1.5) respectively. We have P (z, f) = e2z(e2z + ez + 1) and \delta (0, P ) =
2
4
=
1
2
.
By Theorem 1.3, we can easily get the following two corollaries.
Corollary 1.1. Let f(z) be a transcendental meromorphic function satisfying \sigma 2(f) < 1 and
(1.3). Suppose that P (z, f) is a differential-difference polynomial of the form (1.1) with \mathrm{d}\mathrm{e}\mathrm{g}f P \geq 1.
If P(z, f) contains just one term of maximal total degree and P (z, 0) \not \equiv 0, then \delta (0, P ) = 0.
Corollary 1.2. Let f(z) be a transcendental meromorphic function satisfying \sigma 2(f) < 1 and
(1.3). Suppose that P (z, f) is a homogeneous differential-difference polynomial of the form (1.1)
with \mathrm{d}\mathrm{e}\mathrm{g}f P \geq 1 and P (z, f) \not \equiv 0. Then, for any small meromorphic function \alpha (z) \not \equiv 0 of f(z), we
have \delta (\alpha , P ) = 0.
2. Proof of Theorem 1.1. We need the following lemmas.
Lemma 2.1 [15]. Let f(z) be a transcendental meromorphic function. Let P (f) be a polyno-
mial in f(z) of the form
P (f) = an(z)f(z)
n + an - 1(z)f(z)
n - 1 + . . .+ a1(z)f(z) + a0(z),
where all coefficients aj(z) are small functions of f(z) and an(z) \not \equiv 0. Then
T
\bigl(
r, P (f)
\bigr)
= nT (r, f) + S(r, f).
Lemma 2.2 [6]. Let T : [0,+\infty ) \rightarrow [0,+\infty ) be a nondecreasing continuous function, and let
s \in (0,\infty ). If the hyper order of T is strictly less than one and \delta \in (0, 1 - \sigma 2(T )), then
T (r + s) = T (r) + o(T (r)/r\delta ),
where r runs to infinity outside of a set of finite logarithmic measure.
Let f(z) be a meromorphic function. It is shown in [1] (Lemma 1) and [5, p. 66], that for an
arbitrary c \not = 0, the inequalities\bigl(
1 + o(1)
\bigr)
T
\bigl(
r - | c| , f(z)
\bigr)
\leq T
\bigl(
r, f(z + c)
\bigr)
\leq
\bigl(
1 + o(1)
\bigr)
T
\bigl(
r + | c| , f(z)
\bigr)
hold as r \rightarrow \infty . From its proof we see that the above relations are also true for counting functions.
So by these relations and Lemma 2.2, we get the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
VALUE DISTRIBUTION OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS . . . 475
Lemma 2.3. Let f(z) be a nonconstant meromorphic function of \sigma 2(f) < 1, and let c \not = 0 be
an arbitrary complex number. Then
T
\bigl(
r, f(z + c)
\bigr)
= T (r, f) + S(r, f),
N
\bigl(
r, f(z + c)
\bigr)
= N(r, f) + S(r, f),
N
\bigl(
r, 1/f(z + c)
\bigr)
= N(r, 1/f) + S(r, f).
Applying [6] (Theorem 5.1) to [9] (Theorem 2.3), we get the following lemma.
Lemma 2.4. Let f(z) be a transcendental meromorphic solution of hyper order \sigma 2(f) < 1 of
a difference equation of the form
U(z, f)P (z, f) = Q(z, f),
where U(z, f), P (z, f), Q(z, f) are difference polynomials in f(z) with small meromorphic coeffi-
cients, \mathrm{d}\mathrm{e}\mathrm{g}f U = n and \mathrm{d}\mathrm{e}\mathrm{g}f Q \leq n. Moreover, we assume that U(z, f) contains just one term of
maximal total degree. Then
m
\bigl(
r, P (z, f)
\bigr)
= S(r, f).
Remark 2.1. By a careful inspection of the proof of Lemma 2.4, we see that the same conclusion
holds, if P (z, f), Q(z, f) are differential-difference polynomials in f(z) and the coefficients a\lambda (z)
of P (z, f) and Q(z, f) satisfy m(r, a\lambda ) = S(r, f) instead of T (r, a\lambda ) = S(r, f).
Lemma 2.5 [18]. Let f(z) be a transcendental meromorphic function satisfying \sigma 2(f) < 1 and
N(r, f) = S(r, f). Let P (z, f) be a differential-difference polynomial. Then
T (r, P ) \leq (\mathrm{d}\mathrm{e}\mathrm{g}f P )T (r, f) + S(r, f).
Proof of Theorem 1.1. To prove Theorem 1.1, we follow the main idea in the proof of [17]
(Theorem 1.2). First observe that \psi (z) \not \equiv 0. Indeed, if \psi (z) \equiv 0, then
F (z, f)P1(z, f) \equiv - P2(z, f). (2.1)
Since \mathrm{d}\mathrm{e}\mathrm{g}f P2 < uv = \mathrm{d}\mathrm{e}\mathrm{g}f F, it follows from Lemma 2.4 and Remark 2.1 that
m(r, P1) = S(r, f).
Moreover, Lemma 2.3 and the assumption that N(r, f) = S(r, f) give N(r, P1) = S(r, f). So
T (r, P1) = S(r, f). Therefore, we have from Lemma 2.1 that
T (r, FP1) = uvT (r, f) + S(r, f).
On the other hand, we get from Lemma 2.5 that
T (r, P2) \leq (\mathrm{d}\mathrm{e}\mathrm{g}f P2)T (r, f) + S(r, f). (2.2)
Since \mathrm{d}\mathrm{e}\mathrm{g}f P2 < uv, comparing the characteristic functions of both sides of (2.1), we obtain a
contradiction. So, \psi (z) \not \equiv 0.
Differentiating both sides of
\psi (z) = F (z, f)P1(z, f) + P2(z, f), (2.3)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
476 R. R. ZHANG
we obtain
\psi \prime (z) = F \prime (z, f)P1(z, f) + F (z, f)P \prime
1(z, f) + P \prime
2(z, f). (2.4)
Since \psi (z) \not \equiv 0, multiplying both sides of (2.3) by
\psi \prime (z)
\psi (z)
, we have
\psi \prime (z) =
\psi \prime (z)
\psi (z)
F (z, f)P1(z, f) +
\psi \prime (z)
\psi (z)
P2(z, f). (2.5)
Subtracting (2.5) from (2.4), we get
F \prime (z, f)P1(z, f) + F (z, f)P \prime
1(z, f) -
\psi \prime (z)
\psi (z)
F (z, f)P1(z, f) = P2(z, f)
\biggl(
- P
\prime
2(z, f)
P2(z, f)
+
\psi \prime (z)
\psi (z)
\biggr)
.
(2.6)
Substituting F (z, f) = (av(z)f
v + av - 1(z)f
v - 1 + . . . + a0(z))
u and F \prime (z, f) = u(av(z)f
v +
+ av - 1(z)f
v - 1 + . . .+ a0(z))
u - 1(a\prime v(z)f
v + vav(z)f
v - 1f \prime + . . .+ a\prime 1(z)f + a1(z)f
\prime + a\prime 0(z)) into
(2.6), we obtain
(av(z)f
v + av - 1(z)f
v - 1 + . . .+ a0(z))
u - 1\omega (z) = P2(z, f)
\biggl(
- P
\prime
2(z, f)
P2(z, f)
+
\psi \prime (z)
\psi (z)
\biggr)
, (2.7)
where
\omega (z) = uP1(z, f)(a
\prime
v(z)f
v + vav(z)f
v - 1f \prime + . . .+ a\prime 1(z)f + a1(z)f
\prime + a\prime 0(z))+
+(av(z)f
v + av - 1(z)f
v - 1 + . . .+ a0(z))
\biggl(
P \prime
1(z, f) -
\psi \prime (z)
\psi (z)
P1(z, f)
\biggr)
. (2.8)
We affirm that \omega (z) \not \equiv 0. Otherwise, since P2(z, f) \not \equiv 0, it follows from (2.7) that
\psi \prime (z)
\psi (z)
=
P \prime
2(z, f)
P2(z, f)
.
Integrating this equation, we have
\psi (z) = C1P2(z, f),
where C1 is a nonzero constant. Substituting \psi (z) = C1P2(z, f) into (2.3), we get
F (z, f)P1(z, f) = (C1 - 1)P2(z, f). (2.9)
From (2.9) and F (z, f)P1(z, f) \not \equiv 0, we obtain C1 \not = 1. Using (2.9) and following steps analogous
to (2.1), (2.2), we have a contradiction. Thus, \omega (z) \not \equiv 0.
By (2.7) we get
m(r, (av(z)f
v + av - 1(z)f
v - 1 + . . .+ a0(z))
u - 1) \leq
\leq m
\biggl(
r,
1
\omega
\biggr)
+m(r, P2) +m
\biggl(
r,
P \prime
2
P2
\biggr)
+m
\biggl(
r,
\psi \prime
\psi
\biggr)
+O(1). (2.10)
Next we estimate every term in (2.10). Since \mathrm{d}\mathrm{e}\mathrm{g}f \psi = uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1, by Lemma 2.5, we obtain
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VALUE DISTRIBUTION OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS . . . 477
T (r, \psi ) \leq (uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1)T (r, f) + S(r, f). (2.11)
By (2.2) and (2.11), we see that S(r, \psi ) = S(r, f) and S(r, P2) = S(r, f). So from m
\biggl(
r,
\psi \prime
\psi
\biggr)
=
= S(r, \psi ) and m
\biggl(
r,
P \prime
2
P2
\biggr)
= S(r, P2), we get
m
\biggl(
r,
\psi \prime
\psi
\biggr)
= S(r, f), m
\biggl(
r,
P \prime
2
P2
\biggr)
= S(r, f). (2.12)
By (u - 1)v > \mathrm{d}\mathrm{e}\mathrm{g}f P2 and (2.12), we see that Lemma 2.4 and Remark 2.1 can be applied to
equation (2.7). So we have
m(r, \omega ) = S(r, f).
By (2.8) and N(r, f) = S(r, f), we get
N(r, \omega ) \leq N(r, 1/\psi ) + S(r, f).
Thus,
T (r, \omega ) \leq N(r, 1/\psi ) + S(r, f).
From this inequality and the first main theorem, we obtain
m(r, 1/\omega ) \leq T (r, \omega ) +O(1) \leq N(r, 1/\psi ) + S(r, f). (2.13)
By Lemma 2.1, we have
T (r, (av(z)f
v + av - 1(z)f
v - 1 + . . .+ a0(z))
u - 1) = (u - 1)vT (r, f) + S(r, f).
Noting that N(r, f) = S(r, f), so
m(r, (av(z)f
v + av - 1(z)f
v - 1 + . . .+ a0(z))
u - 1) = (u - 1)vT (r, f) + S(r, f). (2.14)
By (2.2), (2.10), (2.12) – (2.14), we get
(u - 1)vT (r, f) \leq N(r, 1/\psi ) + (\mathrm{d}\mathrm{e}\mathrm{g}f P2)T (r, f) + S(r, f).
Thus,
((u - 1)v - \mathrm{d}\mathrm{e}\mathrm{g}f P2)T (r, f) \leq N(r, 1/\psi ) + S(r, f).
Combining this inequality with (2.11) and noting that (u - 1)v > \mathrm{d}\mathrm{e}\mathrm{g}f P2, we have
\delta (0, \psi ) = 1 -
——–
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow \infty
N(r, 1/\psi )
T (r, \psi )
\leq 1 -
(u - 1)v - \mathrm{d}\mathrm{e}\mathrm{g}f P2
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
=
\mathrm{d}\mathrm{e}\mathrm{g}f P1 + v + \mathrm{d}\mathrm{e}\mathrm{g}f P2
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
< 1.
3. Proof of Theorem 1.2. We need the following lemma.
Lemma 3.1 [11]. Suppose that h is a nonconstant meromorphic function satisfying
N(r, h) +N(r, 1/h) = S(r, h).
Let f = a0h
p + a1h
p - 1 + . . . + ap, and g = b0h
q + b1h
q - 1 + . . . + bq be polynomials in h with
coefficients a0, a1, . . . , ap, b0, b1, . . . , bq being small functions of h and a0b0ap \not \equiv 0. If q \leq p, then
m(r, g/f) = S(r, h).
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478 R. R. ZHANG
Proof of Theorem 1.2. We only discuss the case uv > \mathrm{d}\mathrm{e}\mathrm{g}f P2 as the case P2(z, f) \equiv 0 can be
treated similarly. We first fix some notations for the proof as follows:
P1(z, f) =
\sum
\lambda \in J
a\lambda (z)
\tau \lambda \prod
j=1
\{ f (\alpha \lambda ,j)(z + \delta \lambda ,j)\} \beta \lambda ,j +A(z),
P2(z, f) =
\sum
\mu \in I
b\mu (z)
\sigma \mu \prod
j=1
\{ f (m\mu ,j)(z + \eta \mu ,j)\} n\mu ,j +B(z),
where J and I are index sets, \delta \lambda ,j and \eta \mu ,j are complex constants, \alpha \lambda ,j and m\mu ,j are nonnegative
integers, \beta \lambda ,j and n\mu ,j are positive integers, and the growth orders of a\lambda (z), b\mu (z), A(z) and B(z)
are all less than \sigma (f). Set d(P1) = \mathrm{d}\mathrm{e}\mathrm{g}f P1 and d(P2) = \mathrm{d}\mathrm{e}\mathrm{g}f P2.
Since a and \infty are Borel exceptional values of f(z), by Hadamard’s factorization theorem, we
get
f(z) = H(z)eh(z) + a, (3.1)
where h(z) is a polynomial and H(z) is a meromorphic function such that \sigma (H) < \sigma (f). So
\sigma (f) = \mathrm{d}\mathrm{e}\mathrm{g} h(z) and f(z) is of regular growth. Therefore, a\lambda (z)(\lambda \in J), b\mu (z)(\mu \in I), aj(z),
j = 0, . . . , v, A(z), B(z), H(z), H(z + c) and eh(z+c) - h(z) are all small functions of eh(z) and
N(r, f) = S(r, eh).
Substituting (3.1) into F (z, f), P1(z, f) and P2(z, f), we obtain
F (z, f) = av(z)
uH(z)uveuvh(z) + suv - 1(z)e
(uv - 1)h(z) + . . .+ s1(z)e
h(z)+
+(av(z)a
v + av - 1(z)a
v - 1 + . . .+ a0(z))
u, (3.2)
P1(z, f) = ld(P1)(z)e
d(P1)h(z) + ld(P1) - 1(z)e
(d(P1) - 1)h(z) + . . .+ l1(z)e
h(z)+
+
\sum
\lambda \in J
a\lambda (z)
\tau \lambda \prod
j=1
(a(\alpha \lambda ,j))\beta \lambda ,j +A(z), (3.3)
P2(z, f) = rd(P2)(z)e
d(P2)h(z) + rd(P2) - 1(z)e
(d(P2) - 1)h(z) + . . .+ r1(z)e
h(z)+
+
\sum
\mu \in I
b\mu (z)
\sigma \mu \prod
j=1
(a(m\mu ,j))n\mu ,j +B(z), (3.4)
where F (z, f), P1(z, f) and P2(z, f) are all polynomials of eh(z) and their coefficients are either
small functions of eh(z) or identically zero. Since uv > d(P2), av(z)
uH(z)uv \not \equiv 0 and P1(z, f) \not \equiv 0,
by (3.2) – (3.4), we have
\psi (z) = F (z, f)P1(z, f) + P2(z, f) =
= wq(z)e
qh(z) + wq - 1(z)e
(q - 1)h(z) + . . .+ w1(z)e
h(z) + F (z, a)P1(z, a) + P2(z, a), (3.5)
where wq(z)(\not \equiv 0) is a small function of eh(z), q satisfies uv \leq q \leq uv + d(P1), and wq - 1(z), . . .
. . . , w1(z), F (z, a), P1(z, a) and P2(z, a) are either small functions of eh(z) or identically zero.
Since F (z, a)P1(z, a) + P2(z, a) \not \equiv 0, by Lemma 3.1, we obtain
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VALUE DISTRIBUTION OF DIFFERENTIAL-DIFFERENCE POLYNOMIALS . . . 479
m(r, 1/\psi ) = S(r, eh).
Noting that uv \leq q \leq uv + d(P1), we get from (3.5), the first main theorem and Lemma 2.1 that
N(r, 1/\psi ) = T (r, \psi ) + S(r, eh) \geq uvT (r, eh) + S(r, eh),
T (r, \psi ) \leq (uv + d(P1))T (r, e
h) + S(r, eh).
Thus,
\delta (0, \psi ) \leq
\mathrm{d}\mathrm{e}\mathrm{g}f P1
uv + \mathrm{d}\mathrm{e}\mathrm{g}f P1
< 1.
4. Proof of Theorem 1.3. We rearrange the expression of P (z, f) in the form
P (z, f) =
degf P\sum
l=0
bl(z)f(z)
l,
where for l = 0, . . . ,\mathrm{d}\mathrm{e}\mathrm{g}f P,
bl(z) =
\sum
\lambda \in Jl
a\lambda (z)
\tau \lambda \prod
j=1
\Biggl(
f (\alpha \lambda ,j)(z + \delta \lambda ,j)
f(z)
\Biggr) \beta \lambda ,j
,
and Jl is defined by (1.2). By (1.4) and (1.5), we see that P (z, f) takes the form
P (z, f) =
m\sum
l=k
bl(z)f(z)
l,
where bm(z) \not \equiv 0, bk(z) \not \equiv 0. We see from (1.3), logarithmic derivative lemma and [6] (Theorem
5.1) that the coefficients bl(z), l = k, k + 1, . . . ,m, are all small functions of f(z).
If m > k, then P (z, f) = f(z)kQ(z, f), where Q(z, f) = bm(z)f(z)m - k+ . . .+bk(z). Lemma
2.1, Lemma 3.1, (1.3) and the first main theorem give
N(r, 1/P ) = N(r, 1/Q) + S(r, f) = (m - k)T (r, f) + S(r, f),
T (r, P ) = mT (r, f) + S(r, f).
Therefore, \delta (0, P ) =
k
m
.
If m = k \geq 1, then P (z, f) = bm(z)f(z)m, where bm(z) \not \equiv 0. By (1.3), we easily see that
\delta (0, P ) = 1 =
k
m
.
References
1. Ablowitz M. J., Halburd R. G., Herbst B. On the extension of the Painlev\'e property to difference equations //
Nonlinearity. – 2000. – 13. – P. 889 – 905.
2. Chen Z. X., Shon K. H. Estimates for the zeros of differences of meromorphic functions // Sci. China. Ser. A. – 2009. –
52. – P. 2447 – 2458.
3. Chen Z. X., Yang C. C. On entire solutions of certain type of differential-difference equations // Taiwanese J. Math. –
2014. – 18, № 3. – P. 677 – 685.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
480 R. R. ZHANG
4. Doeringer W. Exceptional values of differential polynomials // Pacif. J. Math. – 1982. – 98. – P. 55 – 62.
5. Gol’dberg A. A., Ostrovskii I. V. Distribution of values of meromorphic functions. – Moscow: Nauka, 1970.
6. Halburd R. G., Korhonen R. J., Tohge K. Holomorphic curves with shift-invariant hyperplane preimages // Trans.
Amer. Math. Soc. – 2014. – 366. – P. 4267 – 4298.
7. Hayman W. K. Meromorphic functions. – Clarendon Press: Oxford, 1964.
8. Korhonen R. A new Clunie type theorem for difference polynomials // J. Difference Equat. and Appl. – 2011. – 17. –
P. 387 – 400.
9. Laine I., Yang C. C. Clunie theorems for difference and q-difference polynomials // J. London Math. Soc. – 2007. –
76. – P. 556 – 566.
10. Lan S. T., Chen Z. X. Relationships between characteristic functions of \Delta nf and f // J. Math. Anal. and Appl. –
2014. – 412. – P. 922 – 942.
11. Li P., Wang W. J. Entire functions that share a small function with its derivative // J. Math. Anal. and Appl. – 2007. –
328. – P. 743 – 751.
12. 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.
13. Liu K., Yang L. Z. On entire solutions of some differential-difference equations // Comput. Methods Funct. Theory. –
2013. – 13. – P. 433 – 447.
14. Liu X. L., Wang L. N., Liu K. The zeros of differential-difference polynomials of certain types // Adv. Difference
Equat. – 2012. – 164. – P. 1 – 8.
15. Yang C. C. On deficiencies of differential polynomials, II // Math. Z. – 1972. – 125. – S. 107–112.
16. Zhang R. R., Chen Z. X. Value distribution of meromorphic functions and their differences // Turk. J. Math. – 2012. –
36. – P. 395 – 406.
17. Zhang R. R., Chen Z. X. Value distribution of difference polynomials of meromorphic functions (in Chinese) // Sci.
Sin. Math. – 2012. – 42, № 11. – P. 1115 – 1130.
18. Zheng X. M., Xu H. Y. On the dificiencies of some differential-difference polynomials // Abstr. Appl. Anal. – 2014. –
Article ID 378151. – P. 1 – 12.
Received 23.04.15,
after revision — 10.12.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
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| id | umjimathkievua-article-1569 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T02:08:15Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f0/e5b20a1fbaee1d8fab7838c0968a3bf0.pdf |
| spelling | umjimathkievua-article-15692019-12-05T09:19:04Z Value distribution of differential-difference polynomials of meromorphic functions Розподiл значень для диференцiально-рiзницевих полiномiв мероморфних функцiй Zhang, R. R. Чжан, Р. Р. We obtain the results on the deficiencies of differential-difference polynomials. These results can be regarded as differential difference analogs of some classical theorems on differential polynomials. In particular, an exact estimate of the deficiency of some differential-difference polynomials is presented. We also give examples showing that these results are best possible in a certain sense. Отримано результати щодо дефектiв диференцiально-рiзницевих полiномiв. Цi результати можна розглядати як диференцiально-рiзницевi аналоги деяких класичних теорем для диференцiальних полiномiв. Зокрема, наведено точну оцiнку для дефектiв деяких диференцiально-рiзницевих полiномiв. Також наведено приклади, якi показують, що цi результати є, в певному сенсi, найкращими. Institute of Mathematics, NAS of Ukraine 2018-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1569 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 4 (2018); 471-480 Український математичний журнал; Том 70 № 4 (2018); 471-480 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/1569/551 Copyright (c) 2018 Zhang R. R. |
| spellingShingle | Zhang, R. R. Чжан, Р. Р. Value distribution of differential-difference polynomials of meromorphic functions |
| title | Value distribution of differential-difference polynomials of meromorphic functions |
| title_alt | Розподiл значень для диференцiально-рiзницевих полiномiв
мероморфних функцiй |
| title_full | Value distribution of differential-difference polynomials of meromorphic functions |
| title_fullStr | Value distribution of differential-difference polynomials of meromorphic functions |
| title_full_unstemmed | Value distribution of differential-difference polynomials of meromorphic functions |
| title_short | Value distribution of differential-difference polynomials of meromorphic functions |
| title_sort | value distribution of differential-difference polynomials of meromorphic functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1569 |
| work_keys_str_mv | AT zhangrr valuedistributionofdifferentialdifferencepolynomialsofmeromorphicfunctions AT čžanrr valuedistributionofdifferentialdifferencepolynomialsofmeromorphicfunctions AT zhangrr rozpodilznačenʹdlâdiferencialʹnoriznicevihpolinomivmeromorfnihfunkcij AT čžanrr rozpodilznačenʹdlâdiferencialʹnoriznicevihpolinomivmeromorfnihfunkcij |