We mainly study the exponents of convergence of zeros and poles of difference and divided difference of transcendental meromorphic solutions for certain difference Painlevé III equations.

1. Introduction and Main Results

In this paper, we use the basic notions of Nevanlinna's theory (see [1, 2]). In addition, we use the notations σ(w) to denote the order of growth of the meromorphic function w(z), λ(w) and λ(1/w), respectively, to denote the exponents of convergence of zeros and poles of w(z). The quantity δ(a,w) is called the deficiency of the value a to w(z). Furthermore, we denote by S(r,w) any quantity satisfying S(r,w)=o(T(r,w)) for all r outside of a set with finite logarithmic measure, and by
(1)𝒮(w)={αmeromorphic:T(r,α)=S(r,w)}
the field of small functions with respect to w. A meromorphic solution w of a difference (or differential) equation is called admissible if all coefficients of the equation are in 𝒮(w).

At the beginning of the last century, Painlevé, Gambier, and Fuchs classified a large number of second order differential equations in terms of a characteristic which is now known as the Painlevé property [3–6]. They are proven to be integrable by using inverse scattering transform technique, for instance [7].

Recently, a number of papers (such as [8–12]) focus on complex difference equations and difference analogues of Nevanlinna's theory. Ablowitz et al. [13] considered discrete equations as delay equations in the complex plane which enabled them to utilize complex analytic methods. They looked at difference equations of the type
(2)w(z+1)+w(z-1)=R(z,w),
where R is rational in both of its arguments. It is shown that if (2) has at least one nonrational finite order meromorphic solution, then degwR≤2.

Recently, Halburd and Korhonen [14] considered (2), where the coefficients of R(z,w) are in 𝒮(w) and got Theorem A.

Theorem A.

If (2) has an admissible meromorphic solution of finite order, where R(z,w) is rational and irreducible in w and meromorphic in z, then either w satisfies a difference Riccati equation
(3)w(z+1)=p(z+1)w(z)+q(z)w(z)+p(z),
where p(z),q(z)∈𝒮(w), or (2) can be transformed to one of the following equations: (4a)w(z+1)+w(z)+w(z-1)=π1z+π2w(z)+κ1,(4b)w(z+1)-w(z)+w(z-1)=π1z+(-1)zπ2w(z)+κ1,(4c)w(z+1)+w(z-1)=π1z+π3w(z)+π2,(4d)w(z+1)+w(z-1)=π1z+κ1w(z)+π2w2(z),(4e)w(z+1)+w(z-1)=π1z+κ1+π2(-1)z-w2(z),(4f)w(z+1)+w(z-1)=π1z+κ1+π21-w2(z),(4g)w(z+1)+w(z-1)=p(z)w(z),(4h)w(z+1)+w(z-1)=p(z)w(z)+q(z),where πk,κk∈𝒮(w) are arbitrary finite order periodic functions with period k.

Equations (4a), (4c), and (4d) are known as difference Painlevé I equations, while (4f) is often viewed as difference Painlevé II equation. Equations (4b) and (4e) are slight variations of (4a) and (4f), respectively.

In 2010, Chen and Shon [15] researched the properties of finite order meromorphic solutions of difference Painlevé I and II equations. They mainly discussed the existence and the forms of rational solutions and value distribution of transcendental meromorphic solutions.

For difference Painlevé III equations, we recall the following.

Theorem B (see [<xref ref-type="bibr" rid="B15">16</xref>]).

Assume that equation
(5)w(z+1)w(z-1)=R(z,w)
has an admissible meromorphic solution w of hyperorder less than one, where R(z,w) is rational and irreducible in w and meromorphic in z; then either w satisfies a difference Riccati equation
(6)w(z+1)=α(z)w(z)+β(z)w(z)+γ(z),
where α(z),β(z),γ(z)∈𝒮(w) are algebroid functions, or (5) can be transformed to one of the following equations: (7a)w(z+1)w(z-1)=η(z)w2(z)-λ(z)w(z)+μ(z)(w(z)-1)(w(z)-ν(z)),(7b)w(z+1)w(z-1)=η(z)w2(z)-λ(z)w(z)w(z)-1,(7c)w(z+1)w(z-1)=η(z)(w(z)-λ(z))w(z)-1,(7d)w(z+1)w(z-1)=h(z)wm(z).In (7a), the coefficients satisfy κ2(z)μ(z+1)μ(z-1)=μ2(z), λ(z+1)μ(z)=κ(z)λ(z-1)μ(z+1), κ(z)λ(z+2)λ(z-1)=κ(z-1)λ(z)λ(z+1), and one of the following:

η≡1,ν(z+1)ν(z-1)=1,κ(z)=ν(z);

η(z+1)=η(z-1)=ν(z),κ≡1.

In (7b), η(z)η(z+1)=1 and λ(z+2)λ(z-1)=λ(z)λ(z+1). In (7c), the coefficients satisfy one of the following:

η≡1 and either λ(z)=λ(z+1)λ(z-1) or λ(z+3)λ(z-3)=λ(z+2)λ(z-2);

Zhang and Yang [17] investigated difference Painlevé III equations (7a)–(7d) with constant coefficients and obtained the following results.

Theorem C.

If w is a nonconstant meromorphic solution of difference equation (7d), where m=-2,-1,0,1 and h is a nonzero constant, then

w cannot be a rational function;

λ(w)=τ(w)=σ(w), where τ(w) denotes the exponent of convergence of fixed points of w.

Theorem D.

If w is a nonconstant meromorphic solution of difference equation (7d), where m=2 and h is a nonzero constant, then

w has no nonzero Nevanlinna exceptional value;

w cannot be a rational function;

τ(w)=σ(w).

In Theorems C and D, h is defined as a nonzero constant. A natural question to ask is what can we say on meromorphic solutions of (7a)–(7d) if h is a nonconstant meromorphic function? In this paper, we answer this question. In the following theorems, we study the properties of difference and divide difference of transcendental meromorphic solutions of (7a)–(7d).

Theorem 1.

Suppose that h(z) is a nonconstant rational function. If w(z) is a transcendental meromorphic solution with finite order of equation
(8)w(z+1)w(z-1)=h(z),
set Δw(z)=w(z+1)-w(z). Then

w has no Nevanlinna exceptional value;

λ(Δw)=λ(1/Δw)=σ(w),λ(Δw/w)=λ(1/(Δw/w))=σ(w).

Example 2.

The function w(z)=z(ei(π/2)z-1)/(ei(π/2)z+1) is a meromorphic solution of difference equation
(9)w(z+1)w(z-1)=(z+1)(z-1),
where h(z)=(z+1)(z-1). By calculation, this solution satisfies
(10)Δw(z)=ieiπz+(i-1)(2z+1)ei(π/2)z-1(iei(π/2)z+1)(ei(π/2)z+1),Δw(z)w(z)=ieiπz+(i-1)(2z+1)ei(π/2)z-1z(iei(π/2)z+1)(ei(π/2)z-1).
Thus,
(11)λ(Δw)=λ(1Δw)=σ(w)=1,λ(Δww)=λ(1Δw/w)=σ(w)=1.

Theorem 3.

Suppose that h(z) is a nonconstant rational function. If w(z) is a transcendental meromorphic solution with finite order of equation
(12)w(z+1)w(z-1)=h(z)w(z),
then

w has no Nevanlinna exceptional value;

λ(Δw)=λ(1/Δw)=σ(w),λ(Δw/w)=λ(1/(Δw/w))=σ(w).

From the following proof of Theorem 3, we have the following.

Remark 4.

If w(z) is an admissible meromorphic solution with finite order of (12), then T(r,Δw/w)=T(r,w)+S(r,w).

Example 5.

The function w(z)=ztan(πz/6)tan((πz/6)-(π/6)) is a meromorphic solution of difference equation
(13)w(z+1)w(z-1)=-z2-1zw(z),
where h(z)=-(z2-1)/z. By calculation, this solution satisfies
(14)Δw(z)=tan(πz6)2sin(πz/3)+3(2z+1)2cos(πz/3)-1,Δw(z)w(z)=cot(πz6-π6)2sin(πz/3)+3(2z+1)z(2cos(πz/3)-1).
Thus,
(15)λ(Δw)=λ(1Δw)=σ(w)=1,λ(Δww)=λ(1Δw/w)=σ(w)=1.

Theorem 6.

Suppose that h(z) is a nonconstant rational function. If w(z) is a transcendental meromorphic solution with finite order of equation
(16)w(z+1)w(z-1)w(z)=h(z),
then

w has no Nevanlinna exceptional value;

λ(Δw)=λ(1/Δw)=σ(w),λ(Δw/w)=λ(1/(Δw/w))=σ(w).

Theorem 7.

Suppose that h(z) is a nonconstant rational function. If w(z) is a transcendental meromorphic solution with finite order of equation
(17)w(z+1)w(z-1)w2(z)=h(z),
then

w has no Nevanlinna exceptional value;

λ(Δw)=λ(1/Δw)=σ(w),λ(Δw/w)=λ(1/(Δw/w))=σ(w).

From the following proof of Theorem 7, we see the following.

Remark 8.

If w(z) is an admissible meromorphic solution with finite order of (17), then T(r,Δw/w)=2T(r,w)+S(r,w).

Example 9.

The function w(z)=z(eiπz-1)/(eiπz+1) is a meromorphic solution of difference equation
(18)w(z+1)w(z-1)w2(z)=z4-z2,
where h(z)=z4-z2. By calculation, this solution satisfies
(19)Δw(z)=ei2πz+(4z+2)eiπz+1ei2πz-1,Δw(z)w(z)=ei2πz+(4z+2)eiπz+1z(eiπz-1)2.
Thus,
(20)λ(Δw)=λ(1Δw)=σ(w)=1,λ(Δww)=λ(1Δw/w)=σ(w)=1.

From the following proofs of Theorems 1–7, we point out the following.

Remark 10.

Suppose that h(z) is a meromorphic function satisfying h(z+1)≢h(z). If w(z) is an admissible meromorphic solution with finite order of (7d), where m=-2,-1,0,1, then Theorems 1–7 still hold.

Equations (7a)–(7c) and w(z+1)w(z-1)=h(z)w2(z) can be discussed similarly; we omit it in the present paper.

2. Lemmas for the Proofs of Theorems Lemma 11 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

Let f(z) be a meromorphic function of finite order and let c be a nonzero complex constant. Then
(21)m(r,f(z+c)f(z))+m(r,f(z)f(z+c))=S(r,f).

Lemma 12 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

Let f(z) be a meromorphic function with order σ=σ(f),σ<∞, and let η be a fixed nonzero complex number, then for each ε>0, we have
(22)T(r,f(z+η))=T(r,f(z))+O(rσ-1+ε)+O(logr).

Lemma 13 (see [<xref ref-type="bibr" rid="B5">9</xref>]).

Let f(z) be a meromorphic function with exponent of convergence of poles λ(1/f)=λ<∞, and let η≠0 be fixed. Then for each ε(0<ε<1),
(23)N(r,f(z+η))=N(r,f(z))+O(rλ-1+ε)+O(logr).

Lemmas 11 and 12 show the following.

Lemma 14.

Let c be a nonzero constant and let f(z) be a finite order meromorphic function. Then
(24)N(r,1f(z+c))=N(r,1f(z))+S(r,f).

Let f(z) be a meromorphic function. Then for all irreducible rational functions in f,
(25)R(z,f(z))=an(z)f(z)n+⋯+a0(z)bm(z)f(z)m+⋯+b0(z)
with meromorphic coefficients ai(z),bj(z) being small with respect to f, the characteristic function of R(z,f(z)) satisfies
(26)T(r,R(z,f(z)))=max{m,n}T(r,f)+S(r,f).

Lemma 16 (see [<xref ref-type="bibr" rid="B8">10</xref>, <xref ref-type="bibr" rid="B11">11</xref>]).

Let w be a transcendental meromorphic solution with finite order of difference equation
(27)P(z,w)=0,
where P(z,w) is a difference polynomial in w(z). If P(z,a)≢0 for a meromorphic function a∈𝒮(w), then
(28)m(r,1w-a)=S(r,w).

Lemma 17 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Let f be a transcendental meromorphic solution with finite order σ of a difference equation of the form
(29)U(z,f)P(z,f)=Q(z,f),
where U(z,f), P(z,f), and Q(z,f) are difference polynomials such that the total degree degfU(z,f)=n in f(z) and its shifts and degfQ(z,f)≤n. If U(z,f) contains just one term of maximal total degree in f(z) and its shifts, then for each ε>0,
(30)m(r,P(z,f))=O(rσ-1+ε)+S(r,f).

3. Proofs of Theorems Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

(i) Set P(z,w):=w(z+1)w(z-1)-h(z)=0. Since h(z) is a nonconstant rational function, for any a∈ℂ, we know P(z,a)=a2-h(z)≢0. Lemma 16 gives m(r,1/(w-a))=S(r,w), which follows N(r,1/(w-a))=T(r,w)+S(r,w). Thus, δ(a,w)=0.

From (8), we have that
(31)w(z+2)w(z)=h(z+1).
Applying Lemma 17 to (31), we know
(32)m(r,w)=S(r,w),
which implies N(r,w)=T(r,w)+S(r,w). Thus, δ(∞,w)=0.

Therefore, for any a∈ℂ∪{∞},δ(a,w)=0. So, w has no Nevanlinna exceptional value.

(ii) First, we prove that λ(Δw/w)=λ(1/(Δw/w))=σ(w). By (8) and Lemma 12, we obtain
(33)2T(r,w(z))=T(r,h(z)w2(z))+O(logr)=T(r,w(z+1)w(z)w(z-1)w(z))+O(logr)≤T(r,w(z+1)w(z))+T(r,w(z)w(z-1))+O(logr)=2T(r,w(z+1)w(z))+S(r,w(z+1)w(z))+O(logr)≤2T(r,w(z+1)w(z))+S(r,w(z)).
Hence,
(34)T(r,w(z))≤T(r,w(z+1)w(z))+S(r,w(z))=T(r,Δw(z)w(z))+S(r,w(z)).
From (34) and Lemmas 11 and 12, we deduce that
(35)N(r,Δww)=T(r,Δww)-m(r,Δww)=T(r,Δww)+S(r,w)≥T(r,w)+S(r,w).
Thus, λ(1/(Δw/w))≥σ(w), that is, λ(1/(Δw/w))=σ(w).

By (8) and (31), we know
(36)w(z+2)w(z+1)w(z)w(z-1)=h(z+1)h(z).

Set
(37)g(z)=w(z+1)w(z).
Thus, (36) can be written as g(z+1)g(z-1)=h(z+1)/h(z). Set P1(z,g):=g(z+1)g(z-1)-(h(z+1)/h(z))=0. Since h(z) is a nonconstant rational function, h(z) cannot be a periodic function. Then P1(z,1)=1-(h(z+1)/h(z))≢0. Since P1(z,1)≢0, by (37) and Lemmas 12 and 16, we have
(38)m(r,1g(z)-1)=S(r,g(z))=S(r,w(z+1)w(z))≤S(r,w(z)).
Thus,
(39)m(r,1Δw(z)/w(z))=m(r,1(w(z+1)/w(z))-1)=m(r,1g(z)-1)=S(r,w(z)).
By (34) and (39), we have
(40)N(r,1Δw/w)=T(r,Δww)-m(r,1Δw/w)=T(r,Δww)+S(r,w)≥T(r,w)+S(r,w).
Then, λ(Δw/w)≥σ(w), that is, λ(Δw/w)=σ(w).

Next, we prove λ(Δw)=λ(1/Δw)=σ(w). By (8),
(41)Δw(z)+Δw(z-1)=(w(z+1)-w(z))+(w(z)-w(z-1))=w(z+1)-w(z-1)=w(z+1)-h(z)w(z+1)=w2(z+1)-h(z)w(z+1).
Applying Lemmas 12 and 15 to (41), we have
(42)2T(r,w(z))=2T(r,w(z+1))+S(r,w(z))=T(r,w2(z+1)-h(z)w(z+1))+S(r,w(z))=T(r,Δw(z)+Δw(z-1))+S(r,w(z))≤T(r,Δw(z))+T(r,Δw(z-1))+S(r,w(z))=2T(r,Δw(z))+S(r,Δw(z))+S(r,w(z))≤2T(r,Δw(z))+S(r,w(z)).
Hence,
(43)T(r,w(z))≤T(r,Δw(z))+S(r,w(z)).

Obviously, it follows from (32) and Lemma 11 that
(44)m(r,Δw)≤m(r,Δww)+m(r,w)=S(r,w).
Together with (43), we have
(45)N(r,Δw)=T(r,Δw)+S(r,w)≥T(r,w)+S(r,w),
which yields λ(1/Δw)≥σ(w). That is, λ(1/Δw)=σ(w).

Set a=0 in (i). By (39), we obtain
(46)m(r,1Δw)=m(r,1Δw/w1w)≤m(r,1Δw/w)+m(r,1w)=S(r,w).
Combining this with (43), we have
(47)N(r,1Δw)=T(r,Δw)-m(r,1Δw)=T(r,Δw)+S(r,w)≥T(r,w)+S(r,w).
Then λ(Δw)≥σ(w), that is, λ(Δw)=σ(w).

Proof of Theorem <xref ref-type="statement" rid="thm1.2">3</xref>.

(i) By (12) and Lemma 11, we see that
(48)m(r,w(z))=m(r,w(z)h(z)w(z)w(z+1)w(z-1))=m(r,h(z)w2(z)w(z+1)w(z-1))=S(r,w),m(r,1w(z))=m(r,1w(z)w(z+1)w(z-1)h(z)w(z))=m(r,w(z+1)w(z-1)h(z)w2(z))=S(r,w).
Hence,
(49)N(r,w(z))=T(r,w(z))+S(r,w),(50)N(r,1w(z))=T(r,w(z))+S(r,w).
So, δ(0,w)=δ(∞,w)=0.

Set
(51)P(z,w):=w(z+1)w(z-1)-h(z)w(z)=0.
Since h(z) is a nonconstant rational function, for any a∈ℂ∖{0}, we have P(z,a)=a2-ah(z)≢0. Lemma 16 gives m(r,1/(w-a))=S(r,w), which follows N(r,1/(w-a))=T(r,w)+S(r,w). Thus, δ(a,w)=0. Combining with δ(0,w)=δ(∞,w)=0, we know w has no Nevanlinna exceptional value.

(ii) First, we prove λ(Δw)=λ(1/Δw)=σ(w). Since (z+1)=w(z)+Δw(z), w(z-1)=w(z)-Δw(z-1), by (12), we have
(52)(w(z)+Δw(z))(w(z)-Δw(z-1))=h(z)w(z);
that is,
(53)(Δw(z)-Δw(z-1))w(z)-Δw(z)Δw(z-1)=-w2(z)+h(z)w(z).

Let z0 be a zero of w(z), not pole of h(z). From (52), z0 is a zero of w(z)+Δw(z) or w(z)-Δw(z-1). Since w(z0)=0, then z0 must be a zero of Δw(z) or Δw(z-1). Thus, by (50) and Lemma 14, we obtain
(54)T(r,w(z))=N(r,1w(z))+S(r,w)≤N(r,1Δw(z))+N(r,1Δw(z-1))+N(r,h(z))+S(r,w)=2N(r,1Δw(z))+S(r,Δw(z))+O(logr)+S(r,w)≤2N(r,1Δw(z))+S(r,w).
Hence, σ(w)≤λ(Δw), that is, λ(Δw)=σ(w).

If z1 is a pole of w(z) with multiplicity k, not pole of h(z), then z1 is a pole of -w2(z)+h(z)w(z) with multiplicity 2k. From (53), one of Δw(z) and Δw(z-1) must have the pole z1 with multiplicity not less than k. Thus, by (49) and Lemma 13, we get
(55)T(r,w(z))=N(r,w(z))+S(r,w)≤N(r,Δw(z))+N(r,Δw(z-1))+N(r,h(z))+S(r,w)=2N(r,Δw(z))+S(r,Δw(z))+O(logr)+S(r,w)≤2N(r,Δw(z))+S(r,w).
Hence, σ(w)≤λ(1/Δw), that is, λ(1/Δw)=σ(w).

Next, we prove that λ(Δw/w)=λ(1/(Δw/w))=σ(w). By (12), we have
(56)Δw(z)w(z)=w(z+1)-w(z)w(z)=w(z+1)w(z)-1=h(z)w(z-1)-1=h(z)-w(z-1)w(z-1).
From (56) and Lemmas 11 and 12, we deduce that
(57)N(r,Δw(z)w(z))=T(r,Δw(z)w(z))-m(r,Δw(z)w(z))=T(r,Δw(z)w(z))+S(r,w(z))=T(r,h(z)w(z-1)-1)+S(r,w(z))=T(r,w(z-1))+S(r,w(z))=T(r,w(z))+S(r,w(z)).
Thus, λ(1/(Δw/w))=σ(w).

Since h(z) is a nonconstant rational function, h(z) cannot be a periodic function. Thus, by (51), P(z,h(z+1))=h(z+2)h(z)-h(z)h(z+1)=h(z)(h(z+2)-h(z+1))≢0. Lemma 16 gives m(r,1/(w(z)-h(z+1)))=S(r,w(z)), which follows
(58)N(r,1w(z)-h(z+1))=T(r,w(z))+S(r,w(z)).
By (56), if z0 is a common zero of h(z)-w(z-1) and w(z-1), then z0 must be a zero of h(z). Thus, by (56), (58), and Lemma 14, we have
(59)N(r,1Δw(z)/w(z))≥N(r,1w(z-1)-h(z))-N(r,1h(z))=N(r,1w(z)-h(z+1))+O(logr)+S(r,w(z))=T(r,w(z))+S(r,w(z)).
Hence, λ(Δw/w)≥σ(w), that is, λ(Δw/w)=σ(w).

Proof of Theorem <xref ref-type="statement" rid="thm1.3">6</xref>.

(i) Set P(z,w):=w(z+1)w(z-1)w(z)-h(z)=0. Since h(z) is a nonconstant rational function, for any a∈ℂ, we have P(z,a)=a3-h(z)≢0. Lemma 16 shows m(r,1/(w-a))=S(r,w), which yields N(r,1/(w-a))=T(r,w)+S(r,w). Thus, δ(a,w)=0.

We see from (16) and Lemma 17 that
(60)m(r,w)=S(r,w),
which follows N(r,w)=T(r,w)+S(r,w); thus, δ(∞,w)=0.

Therefore, for any a∈ℂ∪{∞},δ(a,w)=0. So, w has no Nevanlinna exceptional value.

(ii) First, we prove λ(1/(Δw/w))=λ(1/Δw)=σ(w). By (16) and Lemma 12, we have
(61)3T(r,w(z))=T(r,h(z)w3(z))+O(logr)=T(r,w(z+1)w(z)w(z-1)w(z))+O(logr)≤T(r,w(z+1)w(z))+T(r,w(z)w(z-1))+O(logr)=2T(r,w(z+1)w(z))+S(r,w(z+1)w(z))+O(logr)≤2T(r,w(z+1)w(z))+S(r,w(z)).
Thus,
(62)T(r,Δw(z)w(z))=T(r,w(z+1)w(z))+O(1)≥32T(r,w(z))+S(r,w).
We deduce from (62) and Lemmas 11 and 12 that
(63)N(r,Δww)=T(r,Δww)-m(r,Δww)=T(r,Δww)+S(r,w)≥32T(r,w)+S(r,w).
Then λ(1/(Δw/w))≥σ(w). So, λ(1/(Δw/w))=σ(w).

By (62), we obtain
(64)T(r,Δw)=T(r,Δwww)≥T(r,Δww)-T(r,w)≥12T(r,w)+S(r,w).
By (60), (64), and Lemma 11, we have
(65)N(r,Δw)=T(r,Δw)-m(r,Δw)≥T(r,Δw)-m(r,Δww)-m(r,w)=T(r,Δw)+S(r,w)≥12T(r,w)+S(r,w).
Then λ(1/Δw)≥σ(w), that is, λ(1/Δw)=σ(w).

Next, we prove that λ(Δw)=λ(Δw/w)=σ(w). By (16), we know
(66)w(z+2)w(z)w(z+1)=h(z+1).
By this and (16), we have
(67)w(z+2)w(z+1)w(z)w(z-1)w(z+1)w(z)=h(z+1)h(z).

Set
(68)g(z)=w(z+1)w(z).
Substituting (68) into (67), we have g(z+1)g(z-1)g(z)=h(z+1)/h(z). Set P1(z,g):=g(z+1)g(z-1)g(z)-(h(z+1)/h(z))=0. Since h(z) is a nonconstant rational function, h(z) cannot be a periodic function. Thus, P1(z,1)=1-(h(z+1)/h(z))≢0. By this and by (68) and Lemmas 12 and 16, we obtain
(69)m(r,1g(z)-1)=S(r,g(z))=S(r,w(z+1)w(z))≤S(r,w(z)).
That is,
(70)m(r,1Δw(z)/w(z))=m(r,1(w(z+1)/w(z))-1)=m(r,1g(z)-1)=S(r,w(z)).
By (62) and (70), we have
(71)N(r,1Δw/w)=T(r,Δww)-m(r,1Δw/w)+O(1)=T(r,Δww)+S(r,w)≥32T(r,w)+S(r,w).
Thus, λ(Δw/w)≥σ(w), that is, λ(Δw/w)=σ(w).

Set a=0 in (i). By (70), we have
(72)m(r,1Δw)=m(r,1Δw/w1w)≤m(r,1Δw/w)+m(r,1w)=S(r,w).
Thus, by (64),
(73)N(r,1Δw)=T(r,Δw)-m(r,1Δw)=T(r,Δw)+S(r,w)≥12T(r,w)+S(r,w).
Hence, λ(Δw)≥σ(w)), that is, λ(Δw)=σ(w).

Proof of Theorem <xref ref-type="statement" rid="thm1.4">7</xref>.

The proof of (i) is similar to the proof of (i) in Theorem 6; we omit it here.

(ii) We conclude from (17) and Lemmas 12 and 15 that
(74)4T(r,w(z))=T(r,h(z)w4(z))+O(logr)=T(r,w(z+1)w(z)w(z-1)w(z))+O(logr)≤T(r,w(z+1)w(z))+T(r,w(z)w(z-1))+O(logr)=2T(r,w(z+1)w(z))+S(r,w(z+1)w(z))+O(logr)≤4T(r,w(z))+S(r,w(z)).
Thus,
(75)T(r,Δw(z)w(z))=T(r,w(z+1)w(z))+O(1)=2T(r,w(z))+S(r,w(z)).
By (75) and Lemma 11, we know
(76)N(r,Δww)=T(r,Δww)-m(r,Δww)=T(r,Δww)+S(r,w)=2T(r,w)+S(r,w).
Therefore, λ(1/(Δw/w))=σ(w).

By (17), we know
(77)w(z+2)w(z)w2(z+1)=h(z+1).
By this and (17), we have
(78)w(z+2)w(z+1)w(z)w(z-1)(w(z+1)w(z))2=h(z+1)h(z).

Set
(79)g(z)=w(z+1)w(z).
Then (78) can be written as g(z+1)g(z-1)g2(z)=h(z+1)/h(z). Set P(z,g):=g(z+1)g(z-1)g2(z)-(h(z+1)/h(z))=0. Since h(z) is a nonconstant rational function, h(z) cannot be a periodic function. Thus, P(z,1)=1-(h(z+1)/h(z))≢0. Since P(z,1)≢0, by Lemmas 12 and 16, we have
(80)m(r,1g(z)-1)=S(r,g(z))=S(r,w(z+1)w(z))≤S(r,w(z)),
thus,
(81)m(r,1Δw(z)/w(z))=m(r,1(w(z+1)/w(z))-1)=m(r,1g(z)-1)=S(r,w(z)).
By this and (75), we have
(82)N(r,1Δw/w)=T(r,Δww)-m(r,1Δw/w)=T(r,Δww)+S(r,w)=2T(r,w)+S(r,w).
Then λ(Δw/w)=σ(w).

We see from (76) that
(83)N(r,Δw)=N(r,Δwww)≥N(r,Δww)-N(r,1w)≥N(r,Δww)-T(r,w)=T(r,w)+S(r,w).

We deduce from (82) that
(84)N(r,1Δw)=N(r,1Δw/w1w)≥N(r,1Δw/w)-N(r,w)≥N(r,1Δw/w)-T(r,w)=T(r,w)+S(r,w).
The last two inequalities show λ(1/Δw)≥σ(w) and λ(Δw)≥σ(w), respectively. Thus, λ(Δw)=λ(1/Δw)=σ(w).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The project was supported by the National Natural Science Foundation of China (11171119).

HaymanW. K.YangL.FuchsL.Sur quelques équations différentielles linéares du second ordreGambierB.Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixesPainlevéP.Mémoire sur les équations différentielles dont l'intégrale générale est uniformePainlevéP.Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniformeAblowitzM. J.SegurH.Exact linearization of a Painlevé transcendentChenB. Q.ChenZ. X.LiS.Uniqueness theorems on entire functions and their difference operators or shiftsChiangY.-M.FengS.-J.On the Nevanlinna characteristic of f(z+η) and difference equations in the complex planeHalburdR. G.KorhonenR. J.Difference analogue of the lemma on the logarithmic derivative with applications to difference equationsLaineI.YangC.-C.Clunie theorems for difference and q-difference polynomialsZhangR. R.HuangZ. B.Results on difference analogues of Valiron-Mohon'ko theoremAblowitzM. J.HalburdR.HerbstB.On the extension of the Painlevé property to difference equationsHalburdR. G.KorhonenR. J.Finite-order meromorphic solutions and the discrete Painlevé equationsChenZ.-X.ShonK. H.Value distribution of meromorphic solutions of certain difference Painlevé equationsRonkainenO.Meromorphic solutions of difference Painlevé equationsZhangJ. L.YangL. Z.Meromorphic solutions of Painlevé III difference equationsMokhon’koA. Z.On the Nevanlinna characteristics of some meromorphic functions