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On value distribution and uniqueness of meromorphic function with finite logarithmic order concerning its derivative and qshift difference
Journal of Inequalities and Applications volume 2014, Article number: 295 (2014)
Abstract
In this paper, we study the value distribution of a meromorphic function f(z) concerning its derivative {f}^{\prime}(z) and qshift difference f(qz+c), where f(z) is of finite logarithmic order. We also investigate the uniqueness of differentialqshiftdifference polynomials with more general forms of entire functions of order zero.
1 Introduction and main results
The fundamental theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see e.g. Hayman [1], Yang [2] and Yi and Yang [3]). In addition, for a meromorphic function f(z), we use S(r,f) to denote any quantity satisfying S(r,f)=o(T(r,f)) for all r outside a possible exceptional set E of finite logarithmic measure {lim}_{r\to \mathrm{\infty}}{\int}_{(1,r]\cap E}\frac{dt}{t}<\mathrm{\infty} and also use {S}_{1}(r,f) to denote any quantity satisfying {S}_{1}(r,f)=o(T(r,f)) for all r on a set F of logarithmic density 1, where the logarithmic density of a set F is defined by
The order of a meromorphic function f(z) is defined by
The logarithmic order of a meromorphic function f(z) is defined by (see [4])
If {\rho}_{log}(f)<\mathrm{\infty}, then f(z) is said to be of finite logarithmic order. It is clear that if a meromorphic function f(z) has finite logarithmic order, then f(z) has order zero.
If f(z) is a meromorphic function of finite positive logarithmic order {\rho}_{log}(f), then T(r,f) has proximate logarithmic order {\rho}_{log}(f). The logarithmictype function of T(r,f) is defined as U(r,f)={(logr)}^{{\rho}_{log}(f)}. We have T(r,f)\le U(r,f) for sufficiently large r. The logarithmic exponent of convergence of apoints of f(z) is equal to the logarithmic order of n(r,\frac{1}{fa}), which is defined as
We see by [4] that for a meromorphic function f(z) of finite positive logarithmic order {\rho}_{log}(f), the logarithmic order of N(r,\frac{1}{fa}) is {\lambda}_{log}(a)+1, where {\lambda}_{log}(a) is the logarithmic order of n(r,\frac{1}{fa}).
Moreover, we assume in the whole paper that m, n, k, {t}_{m}, {t}_{n} are positive integers, q\in C\mathrm{\setminus}\{0\}, c\in C, and a(z) is a nonzero small function with respect to f(z), that is, a(z) is a nonzero meromorphic function of growth S(r,f).
Many mathematicians were interested in the value distribution of different expressions of meromorphic functions and obtained lots of important theorems (see e.g. [1, 6–8]). Especially, Hayman [7] discussed Picard’s values of meromorphic functions and their derivatives, and he obtained the following famous theorem in 1959.
Theorem 1.1 [7]
Let f(z) be a transcendental entire function. Then

(a)
for n\ge 2, {f}^{\prime}(z)f{(z)}^{n} assumes all finite values except possibly zero infinitely often;

(b)
for n\ge 3 and a\ne 0, {f}^{\prime}(z)af{(z)}^{n} assumes all finite values infinitely often.
Further, for a transcendental meromorphic function f(z), Chen and Fang [9] obtained the following result.
Theorem 1.2 [[9], Theorem 1]
Let f(z) be a transcendental meromorphic function. If n\ge 1 is a positive integer, {f}^{\prime}(z)f{(z)}^{n}1 has infinitely many zeros.
Recently, with the establishments of difference analogies of the Nevanlinna theory (see e.g. [10–12]), many mathematicians focused on studying difference analogies of Theorems 1.1 and 1.2. The main purpose of these results (see e.g. [5, 13–16]) is to get the sharp estimation of the value of n to make difference polynomials f(z+c)f{(z)}^{n}a and f(z+c)af{(z)}^{n}b admit infinitely many zeros.
Meantime, qdifference analogies of the Nevanlinna theory and their applications on the value distribution of qdifference polynomials and qshiftdifference equations are also studied (see e.g. [17–19]). Especially, for a transcendental meromorphic (resp. entire) function f(z) of order zero, Zhang and Korhonen [20] studied the value distribution of qdifference polynomials of f(z) and found that if n\ge 6 (resp. n\ge 2), then f(qz)f{(z)}^{n} assumes every nonzero value a\in C infinitely often (see [[20], Theorem 4.1]).
Further, Xu and Zhang [21] investigated the zeros of qshift difference polynomials of meromorphic functions of finite logarithmic order and obtained the following result in 2012.
Theorem 1.3 [[21], Theorem 2.1]
If f(z) is a transcendental meromorphic function of finite logarithmic order {\rho}_{log}(f), with the logarithmic exponent of convergence of poles less than {\rho}_{log}(f)1 and q, c are nonzero complex constants, then for n\ge 2, f{(z)}^{n}f(qz+c) assumes every value b\in C infinitely often.
One main aim of this paper is to investigate the zeros of differentialqshiftdifference polynomials about f(z), {f}^{\prime}(z), and f(qz+c), where f(z) is of finite positive logarithmic order.
Theorem 1.4 Let f(z) be a transcendental meromorphic function of finite logarithmic order {\rho}_{log}(f), with the logarithmic exponent of convergence of poles less than {\rho}_{log}(f)1. Set
If n\ge 3, then {F}_{1}(z)a(z) has infinitely many zeros.
We also deal with the value distribution of a differentialqshiftdifference polynomial with another form about f(z), {f}^{\prime}(z) and f(qz+c), and we obtain the following result.
Theorem 1.5 Let f(z) be a transcendental meromorphic function of finite logarithmic order {\rho}_{log}(f), with the logarithmic exponent of convergence of poles less than {\rho}_{log}(f)1. Set
If n\ge 3, then {F}_{2}(z)a has infinitely many zeros, where a\in C.
Some more general differentialqshiftdifference polynomials are investigated in the following.
Theorem 1.6 Let f(z) be a transcendental meromorphic function of finite logarithmic order {\rho}_{log}(f), with the logarithmic exponent of convergence of poles less than {\rho}_{log}(f)1. Set
If m, n satisfy m\ge n+2 or n\ge m+2, then {F}_{3}(z)a(z) has infinitely many zeros.
Let
be a nonzero polynomial, where {a}_{0},{a}_{1},\dots ,{a}_{n} (≠0) are complex constants and {t}_{n} is the number of the distinct zeros of {P}_{n}(z). Then we also obtain the following results.
Theorem 1.7 Let f(z) be a transcendental meromorphic function of finite logarithmic order {\rho}_{log}(f), with the logarithmic exponent of convergence of poles less than {\rho}_{log}(f)1. Set
If m\ge n+k+1, then {F}_{4}(z)a(z) has infinitely many zeros.
Theorem 1.8 Let f(z) be a transcendental meromorphic function of finite logarithmic order {\rho}_{log}(f), with the logarithmic exponent of convergence of poles less than {\rho}_{log}(f)1. Set
If m\ge n+k+1, then {F}_{5}(z)a(z) has infinitely many zeros.
Next, we investigate the uniqueness of differentialqshiftdifference polynomials of entire functions of order zero and obtain the following results.
Theorem 1.9 Let f(z) and g(z) be transcendental entire functions of order zero and n\ge 5. If f{(qz+c)}^{n}{f}^{\prime}(z) and g{(qz+c)}^{n}{g}^{\prime}(z) share a nonzero polynomial p(z) CM, then f{(qz+c)}^{n}{f}^{\prime}(z)=g{(qz+c)}^{n}{g}^{\prime}(z).
Theorem 1.10 Let f(z) and g(z) be transcendental entire functions of order zero and m\ge n+2{t}_{n}+5. If f{(z)}^{m}{P}_{n}(f(qz+c)){f}^{\prime}(z) and g{(z)}^{m}{P}_{n}(g(qz+c)){g}^{\prime}(z) share a nonzero polynomial p(z) CM, then f{(z)}^{m}{P}_{n}(f(qz+c)){f}^{\prime}(z)=g{(z)}^{m}{P}_{n}(g(qz+c)){g}^{\prime}(z).
Theorem 1.11 Let f(z) and g(z) be transcendental entire functions of order zero and n\ge m+2{t}_{m}+5. If {P}_{m}(f(z))f{(qz+c)}^{n}{f}^{\prime}(z) and {P}_{m}(g(z))g{(qz+c)}^{n}{g}^{\prime}(z) share a nonzero polynomial p(z) CM, then {P}_{m}(f(z))f{(qz+c)}^{n}{f}^{\prime}(z)={P}_{m}(g(z))g{(qz+c)}^{n}{g}^{\prime}(z).
2 Some lemmas
To prove the above theorems, we need some lemmas as follows.
Lemma 2.1 [3]
Let f(z) be a nonconstant meromorphic function and P(f)={a}_{0}+{a}_{1}f+\cdots +{a}_{n}{f}^{n}, where {a}_{0},{a}_{1},\dots ,{a}_{n} are complex constants and {a}_{n}\ne 0, then
Lemma 2.2 [21]
Let f(z) be a transcendental meromorphic function of finite logarithmic order and q, η be two nonzero complex constants. Then we have
Lemma 2.3 [[22], Theorem 2.1]
Let f(z) be a nonconstant zeroorder meromorphic function and q\in C\setminus \{0\}. Then
Lemma 2.4 [[3], p.37]
Let f(z) be a nonconstant meromorphic function in the complex plane and l be a positive integer. Then
Lemma 2.5 [4]
If f(z) is a transcendental meromorphic function of finite logarithmic order {\rho}_{log}(f), then for any two distinct small functions a(z) and b(z) with respect to f(z), we have
where U(r,f)={(logr)}^{{\rho}_{log}(f)} is a logarithmictype function of T(r,f). Furthermore, if T(r,f) has a finite lower logarithmic order
with {\rho}_{log}(f)\mu <1, then
Remark 2.1 From the proof of Lemma 2.5 (see [[4], Theorem 7.1]), we can easily see that complex values a and b can be changed into a(z) and b(z), where a(z) and b(z) are two distinct small functions with respect to f(z).
Lemma 2.6 Let f(z) be a transcendental meromorphic function of order zero, {F}_{1}(z)=f{(qz+c)}^{n}{f}^{\prime}(z). Then we have
Proof If f(z) is a meromorphic function of order zero, from Lemmas 2.2 and 2.4, we have
On the other hand, from Lemmas 2.2 and 2.4 again, we have
Thus, we get (1). □
Lemma 2.7 Let f(z) be a transcendental meromorphic function of zero order, {F}_{3}(z)=f{(z)}^{m}f{(qz+c)}^{n}{f}^{\prime}(z). Then we have
and
Proof If f(z) is a meromorphic function of order zero, from Lemmas 2.2 and 2.4, we have
that is, we have (2). On the other hand, from Lemmas 2.2 and 2.4, we have
that is, we have (3), where we assume m\ge n without loss of generality. □
Lemma 2.8 Let f(z) be a transcendental meromorphic function of order zero, {F}_{4}(z)=f{(z)}^{m}{P}_{n}(f(qz+c)){\prod}_{j=1}^{k}{f}^{(j)}(z). Then we have
Proof Since f(z) is a transcendental meromorphic function of order zero, by Lemmas 2.1, 2.2, and 2.4, we can easily get the second inequality. On the other hand, it follows by Lemmas 2.1, 2.2, and 2.4 that
Thus, this completes the proof of Lemma 2.8. □
Similar to Lemma 2.8, we have the following lemma.
Lemma 2.9 Let f(z) be a transcendental meromorphic function of zero order, {F}_{5}(z)={P}_{m}(f(z))f{(qz+c)}^{n}{\prod}_{j=1}^{k}{f}^{(j)}(z). Then we have
3 Proofs of Theorems 1.41.8
3.1 Proof of Theorem 1.4
It follows by Lemma 2.6 that T(r,{F}_{1})=O(T(r,f)) holds for all r on a set of logarithmic density 1. Since f(z) is transcendental and n\ge 3, {F}_{1}(z) is transcendental by Lemma 2.6 again. Since the logarithmic exponent of convergence of poles of f(z) less than {\rho}_{log}(f)1, we have
Assume that {F}_{1}(z)a(z) has only finitely many zeros. Thus, by Lemmas 2.2, 2.42.6, we have
Thus, it follows that
Since n\ge 3, the above inequality implies
which contradicts the fact that T(r,f) has finite logarithmic order {\rho}_{log}(f). Thus, {F}_{1}(z)a(z) has infinitely many zeros, that is, f{(qz+c)}^{n}{f}^{\prime}(z)a(z) has infinitely many zeros.
This completes the proof of Theorem 1.4.
3.2 Proof of Theorem 1.5
Since f(z) is a transcendental meromorphic function of finite logarithmic order, we first claim that {f}^{\prime}(z)+f(z)a\not\equiv 0. In fact, if {f}^{\prime}(z)+f(z)a\equiv 0, that is, \frac{{f}^{\prime}(z)}{f(z)a}\equiv 1. By solving the above equation, we have f(z)=A{e}^{z}+a, where A is a nonzero complex constant. Thus, we have \rho (f)=1, which contradicts the fact that f(z) is of order zero. Thus, set
It follows by Lemmas 2.2 and 2.4 that
that is,
On the other hand, we can easily get
And it follows from (5) and n\ge 3 that
holds for all r on a set of logarithmic density 1. By Lemma 2.2, we have
Assume that f{(qz+c)}^{n}+{f}^{\prime}(z)+f(z)a has finitely many zeros, then
Since the logarithmic exponent of convergence of poles of f(z) is less than {\rho}_{log}(f)1, we have
Then, by Lemmas 2.4, 2.5, and (6), we have
It follows by the above inequality and (4) that
Since n\ge 5, the above inequality implies
which contradicts that T(r,f) has finite logarithmic order {\rho}_{log}(f). Thus, f{(qz+c)}^{n}+{f}^{\prime}(z)+f(z)a has infinitely many zeros.
This completes the proof of Theorem 1.5.
3.3 Proofs of Theorems 1.6, 1.7, and 1.8
Similar to the argument as in Theorem 1.4, by applying Lemmas 2.7, 2.8, and 2.9 instead, we can easily prove Theorems 1.6, 1.7, and 1.8 respectively.
4 Proofs of Theorems 1.91.11
Here, we only give the proof of Theorem 1.10 because the methods of the proofs of Theorems 1.9, 1.10, and 1.11 are very similar.
4.1 Proof of Theorem 1.10
Denote
Since f(z) is a transcendental entire function of order zero, by Lemmas 2.1, 2.2, and 2.4, we have
and
Then it follows from (7) and (8) that
We have by (9) that {S}_{1}(r,F)={S}_{1}(r,f). Similarly, we have {S}_{1}(r,G)={S}_{1}(r,g) and
Since f(z) and g(z) are entire functions of order zero and share p(z) CM, we have
where η is a nonzero constant. If \eta =1, then we have F(z)=G(z), that is, f{(z)}^{m}{P}_{n}(f(qz+c)){f}^{\prime}(z)=g{(z)}^{m}{P}_{n}(g(qz+c)){g}^{\prime}(z).
If \eta \ne 1, then we have
Since {P}_{n}(z) has {t}_{n} distinct zeros, by using the second main theorem and Lemma 2.2, we have
where {\gamma}_{1},{\gamma}_{2},\dots ,{\gamma}_{{t}_{n}} are the distinct zeros of {P}_{n}(z). Similarly, we have
Then (9), (10), (13), and (14) result in
which contradicts m\ge n+2{t}_{n}+5.
This completes the proof of Theorem 1.10.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (11301233, 61202313, 11171119), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001) and Sponsored Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University of China, and the Foundation of Education Department of Jiangxi (GJJ14271, GJJ14644) of China.
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Zheng, XM., Xu, HY. On value distribution and uniqueness of meromorphic function with finite logarithmic order concerning its derivative and qshift difference. J Inequal Appl 2014, 295 (2014). https://doi.org/10.1186/1029242X2014295
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DOI: https://doi.org/10.1186/1029242X2014295
Keywords
 differentialqshiftdifference
 meromorphic function
 logarithmic order