The solutions of one type q-difference functional system

Abstract

In this paper, we study the functional system on q-difference equations, our results can give estimates on the proximity functions and the counting functions of the solutions of q-difference equations system. This implies that solutions have a relatively large number of poles. The main results in this paper concern q-difference equations to the system of q-difference equations.

MSC:30D35, 39B32, 39A13, 39B12.

1 Introduction and main results

A function $f(z)$ is called meromorphic if it is analytic in the complex plane ℂ except at isolate poles. In what follows, we assume that the reader is familiar with the basic notion of Nevanlinna’s value distribution theory, see [1] and [2].

Let us consider the q-difference polynomial case. Let $d_j\in \mathbb{C}$ for $j=1,\dots ,n$, and let $I_q$ be a finite set of multi-indexes $\gamma =({\gamma }_0,\dots ,{\gamma }_n)$. A q-difference polynomial of a meromorphic function $w(z)$ is defined as follows:

$$\begin{array}{rcl}P(z,w)& =& P(z,w(qz),w\left(q^{2}z\right),\dots ,w\left(q^{n}z\right))\\ =& \sum _{\gamma \in I_q}{a}_{\gamma }(z)w{(z)}^{{\gamma }_0}w{(qz)}^{{\gamma }_1}\cdots w{\left(q^{n}z\right)}^{{\gamma }_n},\end{array}$$

(1.1)

where $q\in \mathbb{C}\{0\}$, and the coefficients $a_{\gamma }(z)$ are small meromorphic functions with respect to $w(z)$ such that $T(r,a_{\gamma })=o(T(r,w))$ on a logarithmic density 1, denoted by $S_q(r,w)$. The total degree of $P(z,w)$ in $w(z)$ and the q-shifts of $w(z)$ is denoted by ${deg}_w^q(P)$, and the order of zero of $P(z,x_0,x_1,\dots ,x_n)$, as a function of $x_0$ at $x_0=0$, is denoted as ${ord}_0^q(P)$, which can be found, e.g., in [3]. Moreover, the weight of difference polynomial (1.1) is defined by

$$K_q(P)=\underset{\gamma \in I_q}{max}\left\{\sum _{j=1}^n{\gamma }_j\right\},$$

where γ and $I_q$ are the same as in (1.1) above. The q-difference polynomial $P(z,w)$ is said to be homogeneous with respect to $w(z)$ if the degree $d_{\gamma }={\gamma }_0+\cdots +{\gamma }_n$ of each term in the sum (1.1) is non-zero and the same for all $\gamma \in I_q$.

We recall the following result of Zhang et al. [[4], Theorem 1].

Theorem A Let $w(z)$ be a zero-order meromorphic solution of

$$H(z,w)P(z,w)=Q(z,w),$$

where $P(z,w)$ is a homogeneous q-difference polynomial with polynomial coefficients, and $H(z,w)$ and $Q(z,w)$ are polynomials in $w(z)$ with polynomial coefficients having no common factors. If

$$max\{{deg}_w^q(H),{deg}_w^q(Q)-{deg}_w^q(P)\}>min\{{deg}_w^q(P),{ord}_0^q(Q)\}-{ord}_0^q(P),$$

then $N(r,w)\ne S_q(r,w)$, where ${ord}_0^q(P)$ denotes the order of zero of $P(z,x_0,x_1,\dots ,x_n)$, as a function of $x_0$ at $x_0=0$.

Now let us introduce some notation. Let $q_j\in \mathbb{C}\setminus \{0,\}$ for $j=1,\dots ,n$, and let I and J be a finite set of multi-indexes $I=(i_0,\dots ,i_n)$ and $J=(j_0,\dots ,j_n)$. Two q-difference polynomials of a meromorphic function $w(z)$ are defined as follows:

$$\begin{array}{rcl}{\mathrm{\Omega }}_1(z,w_1,w_2)& =& {\mathrm{\Omega }}_1(z,w_1(z),w_2(z),w_1(q_{1}z),w_2(q_{1}z),\dots ,w_1(q_{n}z),w_2(q_{n}z))\\ =& \sum _{i\in I}{a}_i(z)\prod _{k=1}^2{w}_k{(z)}^{k_{i_0}}{w}_k{(q_{1}z)}^{k_{i_1}}\cdots w_k{(q_{n}z)}^{k_{i_n}}\end{array}$$

and

$$\begin{array}{rcl}{\mathrm{\Omega }}_2(z,w_1,w_2)& =& {\mathrm{\Omega }}_2(z,w_1(z),w_2(z),w_1(q_{1}z),w_2(q_{1}z),\dots ,w_1(q_{n}z),w_2(q_{n}z))\\ =& \sum _{j\in J}{b}_j(z)\prod _{k=1}^2{w}_k{(z)}^{k_{i_0}}{w}_k{(q_{1}z)}^{k_{i_1}}\cdots w_k{(q_{n}z)}^{k_{i_n}},\end{array}$$

where the coefficients $a_i(z)$ and $b_j(z)$ are small with respect to $w_1(z)$ and $w_2(z)$ in the sense that $T(r,a_i)=o(T(r,w_k))$ and $T(r,b_j)=o(T(r,w_k))$, $k=1,2$, on a set of logarithmic density 1, as r tends to infinity outside of an exceptional set E of finite logarithmic measure

$$\underset{r\to \mathrm{\infty }}{lim}{\int }_{E\cap [1,r)}\frac{dt}{t}<\mathrm{\infty }.$$

The weights of ${\mathrm{\Omega }}_1(z,w_1,w_2)$ and ${\mathrm{\Omega }}_2(z,w_1,w_2)$ in $w_1(z)$, $w_2(z)$ are denoted by

$${\lambda }_{11}=\underset{i}{max}\left\{\sum _{l=0}^n{i}_{1l}\right\},{\lambda }_{12}=\underset{i}{max}\left\{\sum _{l=0}^n{i}_{2l}\right\}$$

and

$${\lambda }_{21}=\underset{j}{max}\left\{\sum _{l=0}^n{i}_{1l}\right\},{\lambda }_{22}=\underset{j}{max}\left\{\sum _{l=0}^n{i}_{2l}\right\}.$$

The purpose of this paper is to study the problem of the properties of Nevanlinna counting functions and proximity functions of meromorphic solutions of a type of systems of q-difference equations of the following form:

$$\{\begin{array}{c}{\mathrm{\Omega }}_1(z,w_1,w_2)=R_1(z,w_1),\hfill \\ {\mathrm{\Omega }}_2(z,w_1,w_2)=R_2(z,w_2),\hfill \end{array}$$

(1.2)

where

$$R_1(z,w_1)=\frac{P_1(z,w_1)}{Q_1(z,w_1)}=\frac{{\sum }_{i=0}^{p_1}{a}_i(z)w_1^i}{{\sum }_{j=0}^{q_1}{b}_j(z)w_1^j}$$

and

$$R_2(z,w_2)=\frac{P_2(z,w_2)}{Q_2(z,w_2)}=\frac{{\sum }_{i=0}^{p_2}{c}_i(z)w_2^i}{{\sum }_{j=0}^{q_2}{d}_j(z)w_2^j},$$

the coefficients $\{a_i(z)\}$, $\{b_i(z)\}$, $\{c_i(z)\}$, $\{d_i(z)\}$ are meromorphic functions and small functions. The order of zero of ${\mathrm{\Omega }}_j(z,x_0,\dots ,x_n)$, as a function of $x_0$ at $x_0=0$, is denoted by ${ord}_0({\mathrm{\Omega }}_j)$. The q-difference polynomial ${\mathrm{\Omega }}_k(z,w_1,w_2)$, $k=1,2$, is said to be homogeneous with respect to $w_k(z)$ if the degree $d_k=i_{k0}+\cdots +i_{kn}$ of each term in the sum is non-zero and the same for all $i\in I$. Finally, the order of growth of a meromorphic solution $(w_1,w_2)$ is defined by

$$\rho (w_1,w_2)=max\{\rho (w_1),{\rho }_2(w_2)\},$$

where

$$\rho (w_k)=\underset{r\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{logT(r,w_k)}{logr},k=1,2.$$

In this paper, the main results are as follows.

Theorem 1 Let $(w_1,w_2)$ be a zero-order meromorphic solution of system (1.2), where ${\mathrm{\Omega }}_k(z,w_1,w_2)$ ($k=1,2$) are homogeneous q-difference polynomials in $w_1$ and $w_2$, respectively, with meromorphic coefficients, and $P_k(z,w_k)$ and $Q(z,w_k)$, $k=1,2$, are polynomials in $w_k(z)$ with meromorphic coefficients having no common factors. If

$$max\{q_1,p_1-{\lambda }_{11}\}>min\{{\lambda }_{11},{ord}_{w_1}(P_1)\}-{ord}_{w_1}({\mathrm{\Omega }}_1)+{\lambda }_{12}$$

(1.3)

and

$$max\{q_2,p_2-{\lambda }_{22}\}>min\{{\lambda }_{22},{ord}_{w_2}(P_2)\}-{ord}_{w_2}({\mathrm{\Omega }}_2)+{\lambda }_{21},$$

(1.4)

then $N(r,w_1)=S_q(r,w_1)$ and $N(r,w_2)=S_q(r,w_2)$ cannot hold both at the same time, possibly outside of an exceptional set of finite logarithmic measure.

Theorem 2 Let $(w_1,w_2)$ be a zero-order meromorphic solution of system (1.2), where ${\mathrm{\Omega }}_k(z,w_1,w_2)$ ($k=1,2$) are homogeneous q-difference polynomials in $w_1$ and $w_2$, respectively, with meromorphic coefficients, and $P_k(z,w_k)$ and $Q(z,w_k)$, $k=1,2$, are polynomials in $w_k(z)$ with meromorphic coefficients having no common factors,

$$A=2{\lambda }_{11}-(max\{p_1,q_1+{\lambda }_{11}\}-min\{{\lambda }_{11},{ord}_{w_1}({\mathrm{\Omega }}_1)\})$$

and

$$B=2{\lambda }_{22}-(max\{p_2,q_2+{\lambda }_{22}\}-min\{{\lambda }_{22},{ord}_{w_2}({\mathrm{\Omega }}_2)\}).$$

If $A<0$, $B<0$ and $AB>9{\lambda }_{21}{\lambda }_{12}$, then $m(r,w_k)=S_q(r,w_k)$ ($k=1,2$), where r runs to infinity outside of an exceptional set of finite logarithmic measure.

2 Some lemmas

Lemma 1 ([5], Theorem 1.2)

Let $f(z)$ be a non-constant zero-order meromorphic function, and $q\in \mathbb{C}\setminus \{0\}$. Then

$$m(r,\frac{f(qz)}{f(z)})=S_q(r,f).$$

Lemma 2 ([6], Lemma 4)

If $T:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a piecewise continuous increasing function such that

$$\underset{r\to \mathrm{\infty }}{lim}\frac{logT(r)}{logr}=0,$$

then the set

$$E:=\{r:T(C_{1}r)\ge C_{2}T(r)\}$$

has logarithmic density 0 for all $C_1>1$ and $C_2>1$.

3 Proof of Theorem 1

Since ${\mathrm{\Omega }}_k(z,w_1,w_2)$ are homogeneous in $w_1$ and $w_2$, respectively, it follows by Lemma 1 that

$$m(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{\lambda }_{11}}})\le {\lambda }_{12}m(r,w_2)+S_q(r,w_1)$$

(3.1)

and

$$m(r,\frac{{\mathrm{\Omega }}_2(z,w_1,w_2)}{w_2^{{\lambda }_{22}}})\le {\lambda }_{21}m(r,w_1)+S_q(r,w_2)$$

(3.2)

for all r outside of an exceptional set of finite logarithmic measure. Moreover, from (1.2), we have

$$\begin{array}{rcl}T(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{\lambda }_{11}}})& =& T(r,\frac{P_1(z,w_1)}{Q_1(z,w_1)w_1^{{\lambda }_{11}}})\\ =& (max\{p_1,q_1+{\lambda }_{11}\}-min\{{\lambda }_{11},{ord}_{w_1}(P_1)\})T(r,w_1)\\ +S_q(r,w_1)\end{array}$$

(3.3)

and

$$\begin{array}{rcl}T(r,\frac{{\mathrm{\Omega }}_2(z,w_1,w_2)}{w_2^{{\lambda }_{22}}})& =& T(r,\frac{P_2(z,w_2)}{Q_2(z,w_2)w_2^{{\lambda }_{22}}})\\ =& (max\{p_2,q_2+{\lambda }_{22}\}-min\{{\lambda }_{22},{ord}_{w_2}(P_2)\})T(r,w_2)\\ +S_q(r,w_2),\end{array}$$

(3.4)

where r approaches infinity outside of an exceptional set of finite logarithmic measure. By combining (3.1) and (3.3), (3.2) and (3.4), respectively, it follows that

$$\begin{array}{rcl}N(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{\lambda }_{11}}})& \ge & (1+{\lambda }_{12}+{\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))T(r,w_1)\\ -{\lambda }_{12}m(r,w_2)+S_q(r,w_1)\end{array}$$

(3.5)

and

$$\begin{array}{rcl}N(r,\frac{{\mathrm{\Omega }}_2(z,w_1,w_2)}{w_2^{{\lambda }_{22}}})& \ge & (1+{\lambda }_{21}+{\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))T(r,w_1)\\ -{\lambda }_{21}m(r,w_1)+S_q(r,w_2).\end{array}$$

(3.6)

From Lemma 2, we have

$$\begin{array}{c}N(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{ord}_{w_1}({\mathrm{\Omega }}_1(z,w_1,w_2))}})\hfill \\ \le ({\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))N(qr,w_1)+{\lambda }_{12}N(qr,w_2)+S_q(r,w_1)\hfill \\ =({\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))N(r,w_1)+{\lambda }_{12}N(r,w_2)+S_q(r,w_1)+S_q(r,w_2)\hfill \end{array}$$

and

$$\begin{array}{c}N(r,\frac{{\mathrm{\Omega }}_2(z,w_1,w_2)}{w_1^{{ord}_{w_2}({\mathrm{\Omega }}_2(z,w_1,w_2))}})\hfill \\ \le ({\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))N(qr,w_2)+{\lambda }_{21}N(qr,w_1)+S_q(r,w_2)\hfill \\ =({\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))N(r,w_2)+{\lambda }_{11}N(r,w_1)+S_q(r,w_1)+S_q(r,w_2).\hfill \end{array}$$

Therefore,

$$\begin{array}{rcl}N(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{\lambda }_{11}}})& \le & N(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{ord}_{w_1}({\mathrm{\Omega }}_1(z,w_1,w_2))}})+N(r,\frac{1}{w_1^{{\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1)}})\\ \le & ({\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))N(r,w_1)+{\lambda }_{12}N(r,w_2)\\ +T(r,\frac{1}{w_1^{{\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1)}})+S_q(r,w_1)+S_q(r,w_2)\\ \le & ({\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))N(r,w_1)+{\lambda }_{12}N(r,w_2)\\ +({\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))T(r,w_1)+S_q(r,w_2)+S_q(r,w_2)\end{array}$$

(3.7)

and

$$\begin{array}{rcl}N(r,\frac{{\mathrm{\Omega }}_2(z,w_1,w_2)}{w_2^{{\lambda }_{22}}})& \le & N(r,\frac{{\mathrm{\Omega }}_2(z,w_1,w_2)}{w_2^{{ord}_{w_2}({\mathrm{\Omega }}_2(z,w_1,w_2))}})+N(r,\frac{1}{w_2^{{\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2)}})\\ \le & ({\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))N(r,w_2)+{\lambda }_{21}N(r,w_1)\\ +T(r,\frac{1}{w_2^{{\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2)}})+S_q(r,w_1)+S_q(r,w_2)\\ \le & ({\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))N(r,w_2)+{\lambda }_{21}N(r,w_1)\\ +({\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))T(r,w_2)+S_q(r,w_2)+S_q(r,w_2).\end{array}$$

(3.8)

Combining (3.5) and (3.7), (3.6) and (3.8), respectively, we have

$$\begin{array}{c}(1+{\lambda }_{12}+{\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))T(r,w_1)\hfill \\ <({\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))N(r,w_1)+{\lambda }_{12}T(r,w_2)\hfill \\ +({\lambda }_{11}-{ord}_{w_1}({\mathrm{\Omega }}_1))T(r,w_1)+S_q(r,w_1)+S_q(r,w_2)\hfill \end{array}$$

(3.9)

and

$$\begin{array}{c}(1+{\lambda }_{21}+{\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))T(r,w_2)\hfill \\ <({\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))N(r,w_2)+{\lambda }_{21}T(r,w_1)\hfill \\ +({\lambda }_{22}-{ord}_{w_2}({\mathrm{\Omega }}_2))T(r,w_2)+S_q(r,w_1)+S_q(r,w_2).\hfill \end{array}$$

(3.10)

Suppose that $N(r,w_1)=S_q(r,w_1)$ and $N(r,w_2)=S_q(r,w_2)$, according to (3.9) and (3.10), we have

$$(1+{\lambda }_{12})T(r,w_1)<{\lambda }_{12}T(r,w_2)+S_q(r,w_1)+S_q(r,w_2)$$

and

$$(1+{\lambda }_{21})T(r,w_2)<{\lambda }_{21}T(r,w_1)+S_q(r,w_1)+S_q(r,w_2).$$

That is,

$$(1+{\lambda }_{12}+o(1))T(r,w_1)<({\lambda }_{12}+o(1))T(r,w_2)$$

(3.11)

and

$$(1+{\lambda }_{21}+o(1))T(r,w_2)<({\lambda }_{12}+o(1))T(r,w_1).$$

(3.12)

By (3.11) and (3.12), we conclude that

$$1+{\lambda }_{12}+1+{\lambda }_{21}+o(1)<{\lambda }_{12}+{\lambda }_{21},$$

which is impossible, we prove the assertion.

4 Proof of Theorem 2

It follows by Lemma 1 that

$$m(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{\lambda }_{11}}})\le {\lambda }_{12}m(r,w_2)+S_q(r,w_1)$$

(4.1)

and

$$m(r,\frac{{\mathrm{\Omega }}_2(z,w_1,w_2)}{w_2^{{\lambda }_{22}}})\le {\lambda }_{21}m(r,w_1)+S_q(r,w_2)$$

(4.2)

for all r outside of an exceptional set of finite logarithmic measure.

Suppose now that $(w_1(z),w_2(z))$ is a finite-order meromorphic solution of (1.2). Denoting $C={max}_{j=1,\dots ,n}\{|c_j|\}$ in ${\mathrm{\Omega }}_1(z,w_1,w_2)$ and ${\mathrm{\Omega }}_2(z,w_1,w_2)$, by Lemma 2, we obtain

$$\begin{array}{rcl}N(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{\lambda }_{11}}})& \le & {\lambda }_{11}(N(|q|r,w_1)+N(r,\frac{1}{w_1}))\\ +{\lambda }_{12}(N(|q|r,w_2)+N(r,\frac{1}{w_2}))\\ +{\lambda }_{12}N(r,w_2)+S_q(r,w_1)+S_q(r,w_2)\\ =& {\lambda }_{11}(N(r,w_1)+N(r,\frac{1}{w_1}))+{\lambda }_{12}(N(r,w_2)+N(r,\frac{1}{w_2}))\\ +{\lambda }_{12}N(r,w_2)+S_q(r,w_1)+S_q(r,w_2)\end{array}$$

(4.3)

for all r outside of a set E of finite logarithmic measure. By (4.1) and (4.3), we have

$$\begin{array}{rcl}N(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{\lambda }_{11}}})& \le & {\lambda }_{11}(N(r,w_1)+N(r,\frac{1}{w_1}))\\ +{\lambda }_{12}(N(r,w_2)+N(r,\frac{1}{w_2}))+S_q(r,w_1)+S_q(r,w_2)\\ \le & {\lambda }_{12}(2T(r,w_1)-m(r,w_1))+{\lambda }_{12}(3T(r,w_2)-2m(r,w_2))\\ +S_q(r,w_1)+S_q(r,w_2)\end{array}$$

(4.4)

for all $r\notin E$. On the other hand, by (4.1) and (4.3),

$$\begin{array}{c}N(r,\frac{{\mathrm{\Omega }}_1(z,w_1,w_2)}{w_1^{{\lambda }_{11}}})+{\lambda }_{12}m(r,w_2)\hfill \\ \ge T(r,\frac{P_1(r,w_1)}{w_1^{{\lambda }_{11}}{Q}_{1}r,w_1})\hfill \\ =(max\{p_1,q_1+{\lambda }_{11}\}-min\{{\lambda }_{11},{ord}_{w_1}({\mathrm{\Omega }}_1)\})T(r,w_1)+S_q(r,w_1),\hfill \end{array}$$

(4.5)

where r lies outside of a set F of finite logarithmic measure. Combining inequalities (4.4) and (4.5) with the assumption in Theorem 2, we have

$$\begin{array}{c}(max\{p_1,q_1+{\lambda }_{11}\}-min\{{\lambda }_{11},{ord}_{w_1}({\mathrm{\Omega }}_1)\})T(r,w_1)\hfill \\ -{\lambda }_{12}m(r,w_2)+S_q(r,w_1)+S_q(r,w_2)\hfill \\ \le {\lambda }_{11}(2T(r,w_1)-m(r,w_1))+{\lambda }_{12}(3T(r,w_2)-2m(r,w_2))\hfill \\ +S_q(r,w_1)+S_q(r,w_2).\hfill \end{array}$$

(4.6)

Similarly, we obtain

$$\begin{array}{c}(max\{p_2,q_2+{\lambda }_{22}\}-min\{{\lambda }_{22},{ord}_{w_2}({\mathrm{\Omega }}_2)\})T(r,w_2)\hfill \\ -{\lambda }_{21}m(r,w_1)+S_q(r,w_1)+S_q(r,w_2)\hfill \\ \le {\lambda }_{22}(2T(r,w_2)-m(r,w_2))+{\lambda }_{21}(3T(r,w_1)-2m(r,w_1))\hfill \\ +S_q(r,w_1)+S_q(r,w_2).\hfill \end{array}$$

(4.7)

By (4.6) and (4.7), we obtain

$$\begin{array}{c}{\lambda }_{11}m(r,w_1)\hfill \\ \le (2{\lambda }_{11}-(max\{p_1,q_1+{\lambda }_{11}\}-min\{{\lambda }_{11},{ord}_{w_1}({\mathrm{\Omega }}_1)\})+o(1))T(r,w_1)\hfill \\ +(3{\lambda }_{12}+o(1))T(r,w_2)\hfill \end{array}$$

(4.8)

and

$$\begin{array}{c}((max\{p_2,q_2+{\lambda }_{22}\}-min\{{\lambda }_{22},{ord}_{w_2}({\mathrm{\Omega }}_2)\})-2{\lambda }_{22}+o(1))T(r,w_2)\hfill \\ \le (3{\lambda }_{21}+o(1))T(r,w_1)-2{\lambda }_{21}m(r,w_2).\hfill \end{array}$$

(4.9)

Combining (4.8) and (4.9), we have

$$\begin{array}{c}{\lambda }_{11}m(r,w_1)\hfill \\ \le (2{\lambda }_{11}-(max\{p_1,q_1+{\lambda }_{11}\}-min\{{\lambda }_{11},{ord}_{w_1}({\mathrm{\Omega }}_1)\})+o(1))T(r,w_1)\hfill \\ +\frac{3{\lambda }_{12}(3{\lambda }_{21}+o(1))T(r,w_1)-6{\lambda }_{12}{\lambda }_{21}m(r,w_1)}{(max\{p_2,q_2+{\lambda }_{22}\}-min\{{\lambda }_{22},{ord}_{w_2}({\mathrm{\Omega }}_2)\})-2{\lambda }_{22}},\hfill \end{array}$$

that is,

$$({\lambda }_{11}-\frac{6{\lambda }_{12}{\lambda }_{21}}{B})m(r,w_1)\le (A-\frac{9{\lambda }_{12}{\lambda }_{21}+o(1)}{B})T(r,w_1),$$

(4.10)

where $A=2{\lambda }_{11}-(max\{p_1,q_1+{\lambda }_{11}\}-min\{{\lambda }_{11},{ord}_{w_1}({\mathrm{\Omega }}_1)\})$ and $B=2{\lambda }_{22}-(max\{p_2,q_2+{\lambda }_{22}\}-min\{{\lambda }_{22},{ord}_{w_2}({\mathrm{\Omega }}_2)\})$. Combining the assumption and (4.10), we have

$$m(r,w_1)=S_q(r,w_1)$$

for all r outside of $E\cup F$, a set of finite logarithmic measure.

Similarly, we obtain

$$m(r,w_2)=S_q(r,w_2)$$

for all r outside of $E\cup F$, we have proved the assertion.