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Properties of qshift differencedifferential polynomials of meromorphic functions
Advances in Difference Equations volume 2014, Article number: 249 (2014)
Abstract
In this paper, we deal with the zeros of the qshift differencedifferential polynomials {[P(f){\prod}_{j=1}^{d}f{({q}_{j}z+{c}_{j})}^{{s}_{j}}]}^{(k)}\alpha (z) and {(P(f){\prod}_{j=1}^{d}{[f({q}_{j}z+{c}_{j})f(z)]}^{{s}_{j}})}^{(k)}\alpha (z), where P(f) is a nonzero polynomial of degree n, {q}_{j},{c}_{j}\in \mathbb{C}\setminus \{0\} (j=1,\dots ,d) are constants, n,d,{s}_{j}\phantom{\rule{0.25em}{0ex}}(j=1,\dots ,d)\in {\mathbb{N}}_{+} and \alpha (z) is a small function of f. The results of this paper are an extension of the previous theorems given by Chen and Chen and Qi. We also investigate the value sharing for qshift difference polynomials of entire functions and obtain some results which extend the recent theorem given by Liu, Liu and Cao.
MSC:39A50, 30D35.
1 Introduction and main results
The purpose of this paper is to study some properties of zeros and uniqueness of complex qshift difference polynomials of meromorphic functions. A polynomial {Q}_{q}(z,f) can be called a qshift differencedifferential polynomial in f whenever {Q}_{q}(z,f) is a polynomial in f(z), its qshift f(qz+c) and their derivatives, with small functions of f(z) as the coefficients. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see [1, 2]). A meromorphic function f means f is meromorphic in the complex plane. If no poles occur, then f reduces to an entire function. Given a meromorphic function f(z), a meromorphic function a(z) is called a small function with respect to f if T(r,a(z))=S(r,f), where S(r,f) is used 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}. We 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; the logarithmic density of a set F is defined by
In addition, for some a\in \mathbb{C}\cup \{\mathrm{\infty}\}, if the zeros of f(z)a and g(z)a (if a=\mathrm{\infty}, zeros of f(z)a and g(z)a are the poles of f(z) and g(z), respectively) coincide in locations and multiplicities we say that f(z) and g(z) share the value a CM (counting multiplicities) and if they coincide in locations only we say that f(z) and g(z) share a IM (ignoring multiplicities).
Let l be a nonnegative integer or infinity. For a\in \mathbb{C}\cup \{\mathrm{\infty}\}, we denote by {E}_{l}(a;f) the set of all apoints of f where an apoint of multiplicity k is counted k times if k\le l and l+1 times if k>l. If {E}_{l}(a;f)={E}_{l}(a;g), we say that f, g share the value a with weight l.
Definition 1.2 (see [5])
When f and g share 1 IM, we denote by {\overline{N}}_{L}(r,\frac{1}{f1}) the counting function of the 1points of f whose multiplicities are greater than 1points of g, where each zero is counted only once; similarly, we have {\overline{N}}_{L}(r,\frac{1}{g1}). Let {z}_{0} be a zero of f1 of multiplicity p and a zero of g1 of multiplicity q, we also denote by {N}_{11}(r,\frac{1}{f1}) the counting function of those 1points of f where p=q=1.
In recent years, there has been an increasing interest in studying difference equations, difference products and qdifferences in the complex plane ℂ, and a number of papers (including [6–13]) have focused on the value distribution and uniqueness of differences and differences operator analogs of Nevanlinna theory.
For a transcendental meromorphic function f of finite order, herein and hereinafter, c is a nonzero complex constant and a(z) is small function with respect to f, Liu et al. [14], Chen et al. [15], and Luo and Lin [16] studied the zeros distributions of difference polynomials of meromorphic functions and obtained: if n\ge 6, then f{(z)}^{n}f(z+c)\alpha (z) has infinitely many zeros [[14], Theorem 1.2]; if n>m, then P(f)f(z+c)a(z) has infinitely many zeros (see [[16], Theorem 1]), where P(z)={a}_{n}{z}^{n}+{a}_{n1}{z}^{n1}+\cdots +{a}_{1}z+{a}_{0} is a nonzero polynomial, where {a}_{0},{a}_{1},\dots ,{a}_{n} (≠0) are complex constants, and m is the number of the distinct zeros of P(z).
For transcendental meromorphic (resp. entire) function f of zero order and nonzero complex constant q, Zhang and Korhonen [17] studied the value distribution of qdifference polynomials of meromorphic functions and obtained the result that if n\ge 6 (resp. n\ge 2), then f{(z)}^{n}f(qz) assumes every nonzero value a\in \mathbb{C} infinitely often (see [[17], Theorem 4.1]).
Recently, Liu and Qi [18] firstly investigated the value distributions for qshift of meromorphic function and obtained the following result.
Theorem A (see [[18], Theorem 3.6])
Let f be a zeroorder transcendental meromorphic function, n\ge 6, q\in \mathbb{C}\setminus \{0\}, \eta \in \mathbb{C}, and let R(z) be a rational function. Then the qshift difference polynomial f{(z)}^{n}f(qz+\eta )R(z) has infinitely many zeros.
In this paper, we assume {q}_{j},{c}_{j}\in \mathbb{C}\setminus \{0\} (j=1,\dots ,d) are constants, n,d,{s}_{j}\phantom{\rule{0.25em}{0ex}}(j=1,\dots ,d)\in {\mathbb{N}}_{+}, \lambda ={s}_{1}+{s}_{2}+\cdots +{s}_{d}, and a(z) is a small function of f. We will study the value distribution of difference polynomials of more general form,
where P(f) is a nonzero polynomial of degree n, m is the number of the distinct zeros of P(z), and we obtain the following results.
Theorem 1.1 Let f be a transcendental meromorphic (resp. entire) function of zero order and F(z) be stated as in (1). If k\in \mathbb{N} and n>m(k+1)+2d+1+\lambda (resp. n>m(k+1)+d\lambda). Then {(F(z))}^{(k)}a(z) has infinitely many zeros, where {(F(z))}^{(k)}=F(z), if k=0.
Theorem 1.2 Let f be a transcendental meromorphic (resp. entire) function of zero order and {F}_{1}(z) be stated as in (2). Assume k\in \mathbb{N} and n>(m+2d)(k+1)+\lambda +d+1 (resp. n>(m+2d)(k+1)). Then {({F}_{1}(z))}^{(k)}a(z) has infinitely many zeros, provided that f({q}_{j}z+{c}_{j})\ne f(z), j=1,2,\dots ,d.
Recently, there were obtained some results on the existence and growth of solutions of differencedifferential equations (see [19, 20]). Here, from Theorem 1.1 and Theorem 1.2, we get the following result on some nonlinear qshift differencedifferential equations.
Corollary 1.1 Let p(z), q(z) be nonzero polynomials and F(z) be stated as in (1). Then the nonlinear qshift differencedifferential equation
has no transcendental meromorphic solution of zero order, provided that n\ge \lambda +1.
Corollary 1.2 Let p(z), q(z) be nonzero polynomials and {F}_{1}(z) be stated as in (2). Then the nonlinear qshift differencedifferential equation
has no transcendental meromorphic solution of zero order, provided that f({q}_{j}z+{c}_{j})\ne f(z), j=1,2,\dots ,d, and n\ge \lambda +1.
For the uniqueness of difference and qdifference of meromorphic functions, some results had been obtained (see [17, 21–23]). Here, we only state some of the latest theorems as follows.
Theorem B (see [[16], Theorem 2])
Let f and g be transcendental entire functions of finite order, c be a nonzero complex constant, P(z) be stated as in Theorem 1.3, and let n>2{\mathrm{\Gamma}}_{0}+1 be an integer, where {\mathrm{\Gamma}}_{0}={m}_{1}+2{m}_{2}, {m}_{1} is the number of the simple zero of P(z), and {m}_{2} is the number of multiple zeros of P(z). If P(f)f(z+c) and P(g)g(z+c) share 1 CM, then one of the following results holds:

(i)
f\equiv tg for a constant t such that {t}^{l}=1, where l=GCD\{{\lambda}_{0},{\lambda}_{1},\dots ,{\lambda}_{n}\} and
{\lambda}_{i}=\{\begin{array}{ll}i+1,& {a}_{i}\ne 0,\\ n+1,& {a}_{i}=0,\end{array}\phantom{\rule{1em}{0ex}}i=0,1,2,\dots ,n. 
(ii)
f and g satisfy the algebraic equation R(f,g)\equiv 0, where R({\omega}_{1},{\omega}_{2})=P({\omega}_{1}){\omega}_{1}(z+c)P({\omega}_{2}){\omega}_{2}(z+c);

(iii)
f(z)={e}^{\alpha (z)}, g(z)={e}^{\beta (z)}, where \alpha (z) and \beta (z) are two polynomials, b is a constant satisfying \alpha +\beta \equiv b and {a}_{n}^{2}{e}^{(n+1)b}=1.
In this paper, we will investigated the uniqueness problem of qshifts of entire functions and obtain the following results.
Theorem 1.3 Let f, g be transcendental entire functions of zero order, F(z) be stated as in (1) and
where {\mathrm{\Gamma}}_{0} is stated as in Theorem B. If F(z) and G(z) share 1 CM and n>max\{2({\mathrm{\Gamma}}_{0}+2d)\lambda ,\lambda \}, then one of the following cases holds:

(i)
f\equiv tg for a constant t such that {t}^{\kappa}=1 where \kappa =GCD\{{\lambda}_{0}+\lambda ,{\lambda}_{1}+\lambda ,\dots ,{\lambda}_{n}+\lambda \} and
{\lambda}_{i}=\{\begin{array}{ll}i+1,& {a}_{i}\ne 0,\\ n+1,& {a}_{i}=0,\end{array}\phantom{\rule{1em}{0ex}}i=0,1,2,\dots ,n. 
(ii)
f and g satisfy the algebraic equation R(f,g)\equiv 0, where
R({\omega}_{1},{\omega}_{2})=P({\omega}_{1})\prod _{j=1}^{d}{\omega}_{1}{({q}_{j}z+{c}_{j})}^{{s}_{j}}P({\omega}_{2})\prod _{j=1}^{d}{\omega}_{2}{({q}_{j}z+{c}_{j})}^{{s}_{j}}.
Theorem 1.4 Under the assumptions of Theorem 1.3, if
and l, n, m are integers satisfying one of the following conditions:

(I)
l\ge 3, n>max\{2{\mathrm{\Gamma}}_{0}+4d\lambda ,\lambda \};

(II)
l=2, n>max\{2{\mathrm{\Gamma}}_{0}+5d+m\lambda d\chi ,\lambda \};

(III)
l=1, n>max\{2{\mathrm{\Gamma}}_{0}+6d+2m\lambda 2d\chi ,\lambda \};

(IV)
l=0, n>max\{2{\mathrm{\Gamma}}_{0}+7d+3m\lambda 3d\chi ,\lambda \}.
Then the conclusions of Theorem 1.3 hold, where \chi =min\{\mathrm{\Theta}(0,f),\mathrm{\Theta}(0,g)\}.
2 Some lemmas
In the following, we explain some definitions and notations which are used in this paper. For a\in \mathbb{C}\cup \{\mathrm{\infty}\}, we define
For a\in \mathbb{C}\cup \{\mathrm{\infty}\} and k is a positive integer, we denote by {\overline{N}}_{(k}(r,\frac{1}{fa}) the counting function of those apoints of f whose multiplicities are not less than k in counting the apoints of f we ignore the multiplicities (see [1]) and {N}_{k}(r,\frac{1}{fa})=\overline{N}(r,\frac{1}{fa})+{\overline{N}}_{(2}(r,\frac{1}{fa})+\cdots +{\overline{N}}_{(k}(r,\frac{1}{fa}).
Lemma 2.1 (see [2])
Let f and g be two meromorphic functions. If f and g share 1 CM, then one of the following three cases holds:

(i)
T(r,f)+T(r,g)\le 2{N}_{2}(r,f)+2{N}_{2}(r,g)+2{N}_{2}(r,\frac{1}{f})+2{N}_{2}(r,\frac{1}{g})+S(r,f)+S(r,g);

(ii)
f\equiv g;

(iii)
f\cdot g=1.
Lemma 2.2 (see [24])
Let f and g be two meromorphic functions, and let l be a positive integer. If {E}_{l}(1;f)={E}_{l}(1;g), then one of the following cases must occur:

(i)
\begin{array}{rl}T(r,f)+T(r,g)\le & {N}_{2}(r,f)+{N}_{2}(r,g)+{N}_{2}(r,\frac{1}{f})+{N}_{2}(r,\frac{1}{g})+\overline{N}(r,\frac{1}{f1})\\ +\overline{N}(r,\frac{1}{g1}){N}_{11}(r,\frac{1}{f1})+{\overline{N}}_{(l+1}(r,\frac{1}{f1})\\ +{\overline{N}}_{(l+1}(r,\frac{1}{g1})+S(r,f)+S(r,g);\end{array}

(ii)
f=\frac{(b+1)g+(ab1)}{bg+(ab)}, where a (≠0), b are two constants.
Lemma 2.3 (see [24])
Let f and g be two meromorphic functions. If f and g share 1 IM, then one of the following cases must occur:

(i)
\begin{array}{rl}T(r,f)+T(r,g)\le & 2[{N}_{2}(r,f)+{N}_{2}(r,\frac{1}{f})+{N}_{2}(r,g)+{N}_{2}(r,\frac{1}{g})]\\ +3{\overline{N}}_{L}(r,\frac{1}{f1})+3{\overline{N}}_{L}(r,\frac{1}{g1})+S(r,f)+S(r,g);\end{array}

(ii)
f=\frac{(b+1)g+(ab1)}{bg+(ab)}, where a (≠0), b are two constants.
By combining [7] and [17], we get the following lemma easily.
Lemma 2.4 Let f(z) be a transcendental meromorphic function of zero order and q, η be two nonzero complex constants. Then
Lemma 2.5 (see [[18], Theorem 2.1])
Let f(z) be a nonconstant zeroorder meromorphic function and q\in \mathbb{C}\setminus \{0\}. Then
on a set of logarithmic density 1.
Lemma 2.6 Let f be a transcendental meromorphic function of zero order, and F(z) be stated as in (1). Then we have
If f is a transcendental entire function of zero order, we have
where \lambda ={s}_{1}+{s}_{2}+\cdots +{s}_{d}.
Proof If f is a transcendental entire function of zero order, from the ValironMohon’ko lemma and Lemma 2.5, we have
On the other hand, from Lemma 2.5, we have
Thus, we get (6).
If f is a meromorphic function of zero order, from the ValironMohon’ko lemma and Lemma 2.4, we have
On the other hand, from the ValironMo’honko lemma and Lemma 2.5, we have
Thus, we get (5). □
Using the same method as in Lemma 2.6, we get the following lemma easily.
Lemma 2.7 Let f be a transcendental meromorphic function of zero order, and {F}_{1}(z) be stated as in (2). Then we have
If f is a transcendental entire function of zero order, we have
Lemma 2.8 (see [2] and [[25], Lemma 2.4])
Let f be a nonconstant meromorphic function, and p, k be positive integers. Then
Lemma 2.9 Let f(z) and g(z) be transcendental entire functions of zero order, P(z) be stated as in Theorem 1.1. If n>\lambda, then for any complex constant t\ne 0, we have
Proof For any complex constant t\ne 0, suppose that
Suppose that the roots of P(z)=0 are {b}_{1},{b}_{2},\dots ,{b}_{m} with multiplicities {l}_{1},{l}_{2},\dots ,{l}_{m}. Then we have {l}_{1}+{l}_{2}+\cdots +{l}_{m}=n. From (7), we have
Since f, g are nonconstant entire functions, from (8), we deduce that {b}_{1}={b}_{2}=\cdots ={b}_{m}=0. If fact, from (8), we get that {b}_{1},{b}_{2},\dots ,{b}_{m} are the Picard exceptional values. If m\ge 2 and {b}_{j}\ne 0 (j=1,2,\dots ,m), by Picard’s theorem of entire function, we can see that the Picard exceptional values of f are at least three. Thus, we get a contradiction. Hence, m=1 and {l}_{1}=n, that is, there exists a complex constant γ satisfying P(f)={a}_{n}{(f\gamma )}^{n} and P(g)={a}_{n}{(g\gamma )}^{n}. Then
Since f, g are transcendental entire functions, by the Picard theorem, we can see that f\gamma =0 and g\gamma =0 do not have zeros. Then we obtain f(z)={e}^{\alpha (z)}+\gamma, g(z)={e}^{\beta (z)}+\gamma where \alpha (z), \beta (z) are two nonconstant functions. From (9), we see that f({q}_{j}z+{c}_{j})\ne 0 and g({q}_{j}z+{c}_{j})\ne 0. Thus, we get \gamma =0, that is,
Set M(z)=f(z)g(z). If M(z) is nonconstant, from (10), we have
that is,
Since f, g are transcendental entire functions of zero order, from (11), Lemma 2.4 and n>\lambda, we get a contradiction.
Thus, M(z) is a constant. From (11), we get f(z)g(z)\equiv \mu, where μ is a complex constant satisfying {a}_{n}^{2}{\mu}^{n+\lambda}\equiv t. Since f, g are entire functions of zero order, then f, g are constants, which is a contradiction with f, g being transcendental. Hence,
This completes the proof of Lemma 2.9. □
3 Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1 From (1), by the ValironMohon’ko lemma and Lemma 2.6, we find that F(z) is not constant and {S}_{1}(r,{F}^{(k)})={S}_{1}(r,F)={S}_{1}(r,f). Next, we will consider the following two cases when k\ge 1.
Case 1. If f is a transcendental meromorphic function of zero order, we first suppose that {(P(f){\prod}_{j=1}^{d}f{({q}_{j}z+{c}_{j})}^{{s}_{j}})}^{(k)}=a(z) has finitely solutions. By the Second Fundamental Theorem for three small functions (see [[1], Theorem 2.25]) and the ValironMohon’ko lemma, we have
By Lemma 2.6 and Lemma 2.8, we obtain
From the definitions of {S}_{1}(r,f), S(r,f) and n>m(k+1)+2d+1+\lambda, we get a contradiction to (12). Then F{(z)}^{(k)}a(z) has infinitely many zeros.
Case 2. If f is a transcendental entire function. Suppose that {(P(f){\prod}_{j=1}^{d}f{({q}_{j}z+{c}_{j})}^{{s}_{j}})}^{(k)}=a(z) has finitely solutions. By using the same argument as in Case 1 and (4), we have
which is a contradiction with n>m(k+1)+d\lambda.
For k=0, similar to the proofs of Case 1 and Case 2, and by the Second Fundamental Theorem and Lemma 2.6, we get the conclusions of Theorem 1.1.
Thus, we complete the proof of Theorem 1.1. □
Proof of Theorem 1.2 Similar to the proof of Theorem 1.1, and using Lemma 2.8, we can prove Theorem 1.2 easily. □
4 Proofs of Corollaries 1.1 and 1.2
The proofs of Corollaries 1.1 and 1.2 are similar. Here, we just give the proof of Corollary 1.2.
Proof of Corollary 1.2 Assume that f is a transcendental meromorphic solution of zero order of (4), provided that f({q}_{j}z+{c}_{j})\ne f(z), j=1,2,\dots ,d, then
Integrating above equation k times, it follows
where H(z) is a polynomial. From Lemma 2.5 and since f is a meromorphic function of zero order, we have
which is contradiction with n>\lambda.
This completes the proof of Corollary 1.2. □
5 Proofs of Theorems 1.3 and 1.4
Proof of Theorem 1.3 From the assumptions of Theorem 1.3, we see that F(z), G(z) share 1 CM. Then the following three cases will be considered.
Case 1. Suppose that F(z), G(z) satisfy Lemma 2.1(i). Since f(z), g(z) are entire functions of zero order, from the ValironMohon’ko lemma and Lemma 2.6, we have {S}_{1}(r,F)={S}_{1}(r,f), {S}_{1}(r,G)={S}_{1}(r,g),
and
Then, from Lemma 2.1(i) and Lemma 2.7, since f, g are entire, we have
From Lemma 2.6 and (12), we have
that is,
Since n>2({\mathrm{\Gamma}}_{0}+2d)\lambda and f, g are transcendental functions, we get a contradiction.
Case 2. If F(z)\equiv G(z), that is,
Set h=\frac{f}{g}. If h is not a constant, from (17), we find that f and g satisfy the algebraic equation R(f,g)\equiv 0, where
If h is a constant. Substituting f=gh into (17), we get
where {a}_{n}\phantom{\rule{0.25em}{0ex}}(\ne 0),{a}_{n1},\dots ,{a}_{0} are constants.
Since g is transcendental entire function, we have {\prod}_{j=1}^{d}g{({q}_{j}z+{c}_{j})}^{{s}_{j}}\not\equiv 0. Then, from (18), we have
If {a}_{n}\ne 0 and {a}_{n1}={a}_{n2}=\cdots ={a}_{0}=0, then from (19) and g is transcendental function, we get {h}^{n+\lambda}=1.
If {a}_{n}\ne 0 and there exists {a}_{i}\ne 0 (i\in \{0,1,2,\dots ,n1\}). Suppose that {h}^{n+\lambda}\ne 1, from (19), we have T(r,g)=S(r,g) which is contradiction with transcendental function g. Then {h}^{n+\lambda}=1. Similar to this discussion, we can see that {h}^{j+\lambda}=1 when {a}_{j}\ne 0 for some j=0,1,\dots ,n.
Thus, from the definition of l, we can see that f\equiv tg where t is a constant such that {t}^{\kappa}=1, \kappa =GCD\{{\lambda}_{0}+\lambda ,{\lambda}_{1}+\lambda ,\dots ,{\lambda}_{n}+\lambda \}.
Case 3. If F(z)G(z)\equiv 1. From Lemma 2.9, we get that fg=\mu for a constant μ such that {a}_{n}^{2}{\mu}^{n+\lambda}\equiv 1.
Thus, this completes the proof of Theorem 1.3. □
Proof of Theorem 1.4 From the assumptions of Theorem 1.4, we have {E}_{l}(1;F(z))={E}_{l}(1;G(z)).

(I)
l\ge 3. Since
\begin{array}{r}\overline{N}(r,\frac{1}{F(z)1})+\overline{N}(r,\frac{1}{G(z)1})+{\overline{N}}_{(l+1}(r,\frac{1}{F(z)1})\\ \phantom{\rule{2em}{0ex}}+{\overline{N}}_{(l+1}(r,\frac{1}{G(z)1}){N}_{11}(r,\frac{1}{F(z)1})\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}N(r,\frac{1}{F(z)1})+\frac{1}{2}(r,\frac{1}{G(z)1})+S(r,F)+S(r,G)\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}T(r,F)+\frac{1}{2}T(r,G)+S(r,F)+S(r,G).\end{array}
Case 1. Suppose that F(z), G(z) satisfy Lemma 2.2(i). From (13), (14), and Lemma 2.6, we have
that is,
Since n>2{\mathrm{\Gamma}}_{0}+4d\lambda and f, g are transcendental, a contradiction is obtained.
Case 2. If F(z), G(z) satisfy Lemma 2.2(ii), that is,
where a (≠0), b are two constants.
We now will consider three subcases as follows.
Subcase 2.1. b\ne 0,1. If ab1\ne 0, then by (20) we know
Since f, g are entire functions of zero order, by the Second Fundamental Theorem, we have
Then from Lemma 2.6, we have
Similarly, we have
From the definitions of m and {\mathrm{\Gamma}}_{0}, we have m={m}_{1}+{m}_{2}. Since n>2{\mathrm{\Gamma}}_{0}+4d\lambda, we have n+{\lambda}_{2}m2d>2{\mathrm{\Gamma}}_{0}+4d\lambda +\lambda 2m2d=2{\mathrm{\Gamma}}_{0}+2d2m>0. From the above two inequalities, for any ε (0<\epsilon <2{\mathrm{\Gamma}}_{0}+2d2m), we have
which is a contradiction with f, g are transcendental.
If ab1=0, then by (20) we know F=((b+1)G)/(bG+1). Since f, g are entire functions, we see that \frac{1}{b} is a Picard’s exceptional value of G(z). By the Second Fundamental Theorem, we have
Then, from Lemma 2.6 and n>2{\mathrm{\Gamma}}_{0}+4d\lambda, we know T(r,g)\le {S}_{1}(r,g), a contradiction.
Subcase 2.2. b=1. Then (20) becomes F=a/(a+1G).
If a+1\ne 0, then a+1 is a Picard exceptional value of G. Similar to the discussion as in Subcase 2.1, we can deduce a contradiction again.
If a+1=0, then FG\equiv 1, that is,
Since n>max\{2{\mathrm{\Gamma}}_{0}+4d\lambda ,\lambda \}, by Lemma 2.9, we see that fg=\mu for a constant μ such that {a}_{n}^{2}{\mu}^{n+\lambda}\equiv 1.
Subcase 2.3. b=0. Then (20) becomes F=(G+a1)/a.
If a1\ne 0, then \overline{N}(r,\frac{1}{G+a1})=\overline{N}(r,\frac{1}{F}). Similar to the discussion as in Subcase 2.1, we can deduce a contradiction again.
If a1=0, then F\equiv G, that is,
Using the same argument as in the proof of Case 2 in Theorem 1.3, we can see that f, g satisfy Theorem 1.3(ii).

(II)
l=2. Since
\begin{array}{r}\overline{N}(r,\frac{1}{F1})+\overline{N}(r,\frac{1}{G1}){N}_{11}(r,\frac{1}{F1})\\ \phantom{\rule{2em}{0ex}}+\frac{1}{2}{\overline{N}}_{(l+1}(r,\frac{1}{F1})+\frac{1}{2}{\overline{N}}_{(l+1}(r,\frac{1}{G1})\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}N(r,\frac{1}{F1})+\frac{1}{2}N(r,\frac{1}{G1})\\ \phantom{\rule{1em}{0ex}}\le \frac{1}{2}T(r,F)+\frac{1}{2}T(r,G)+S(r,F)+S(r,G),\end{array}(21)
and
Case 1. If F(z), G(z) satisfy Lemma 2.2(i), from the fact that f(z), g(z) are transcendental entire functions and (21)(22), we have
From (13), (14), Lemma 2.6, and \chi =min\{\mathrm{\Theta}(0,f),\mathrm{\Theta}(0,g)\}, for any ε (0<\epsilon <n+\lambda m2{\mathrm{\Gamma}}_{0}5d+d\chi), we have
Since n>2{\mathrm{\Gamma}}_{0}+5d+m\lambda d\chi and f, g are transcendental functions, we get a contradiction.
Case 2. If F(z), G(z) satisfy Lemma 2.2(ii). Similar to the proof of Case 2 in (I), we get the conclusions of Theorem 1.4.

(III)
l=1. Since
\begin{array}{rl}\overline{N}& (r,\frac{1}{F1})+\overline{N}(r,\frac{1}{G1}){N}_{11}(r,\frac{1}{F1})\\ \le \frac{1}{2}N(r,\frac{1}{F1})+\frac{1}{2}N(r,\frac{1}{G1})\\ \le \frac{1}{2}T(r,F)+\frac{1}{2}T(r,G)+S(r,F)+S(r,G),\end{array}(24)
we have
and
Case 1. If F(z), G(z) satisfy Lemma 2.2(i), from f, g are entire functions, (13), (14), (21), (22), (23), and (24), we have
From Lemma 2.6 and \chi =min\{\mathrm{\Theta}(0,f),\mathrm{\Theta}(0,g)\}, for any ε (0<\epsilon <n+\lambda 2{\mathrm{\Gamma}}_{0}6d2m+2d\chi), we have
Since n>2{\mathrm{\Gamma}}_{0}+6d+2m\lambda 2d\chi, from (27) and f, g are transcendental, we get a contradiction.
Case 2. If F(z), G(z) satisfy Lemma 2.2(ii). Similar to the proof of Case 2 in (I), we get the conclusions of Theorem 1.4.

(IV)
l=0, that is, F(z), G(z) share 1 IM. From the definitions of F(z), G(z), we have
\begin{array}{rl}{\overline{N}}_{L}(r,\frac{1}{F1})& \le N(r,\frac{F}{{F}^{\prime}})=N(r,\frac{{F}^{\prime}}{F})+S(r,F)\le \overline{N}(r,\frac{1}{F})+S(r,F)\\ \le mT(r,f)+d\overline{N}(r,\frac{1}{f})+{S}_{1}(r,f),\end{array}(28)
similarly, we have
Case 1. Suppose that F(z), G(z) satisfy Lemma 2.3(i). From (28) and (29), we have
From Lemma 2.6 and (13)(14), for any ε (0<\epsilon <n+\lambda 2{\mathrm{\Gamma}}_{0}7d3m+3d\chi), we get
Since n>2{\mathrm{\Gamma}}_{0}+7d+3m\lambda 3d\chi, we get a contradiction.
Case 2. Suppose that F(z), G(z) satisfy Lemma 2.3(ii). Similar to the proof of Case 2 in (I), we get the conclusions of Theorem 1.4 easily.
Thus, the proof of Theorem 1.4 is completed. □
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Acknowledgements
The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (11301233, 61202313), the Natural Science Foundation of JiangXi Province in China (20132BAB211001), the Foundation of Education Department of Jiangxi (GJJ14644) of China, and the Humanities and Social Sciences of the Chinese Education Ministry (13YJA760064).
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HYX completed the main part of this article, XLW, HYX and TSZ corrected the main theorems. All authors read and approved the final manuscript.
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Wang, XL., Xu, HY. & Zhan, TS. Properties of qshift differencedifferential polynomials of meromorphic functions. Adv Differ Equ 2014, 249 (2014). https://doi.org/10.1186/168718472014249
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DOI: https://doi.org/10.1186/168718472014249