AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 208701 10.1155/2014/208701 208701 Research Article On Properties of Meromorphic Solutions of Certain Difference Painlevé III Equations Lan Shuang-Ting http://orcid.org/0000-0003-3948-8325 Chen Zong-Xuan Shon Kwang Ho School of Mathematical Sciences South China Normal University Guangzhou 510631 China scnu.edu.cn 2014 2722014 2014 05 11 2013 13 01 2014 27 2 2014 2014 Copyright © 2014 Shuang-Ting Lan and Zong-Xuan Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ) = { α    meromorphic : 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 . They are proven to be integrable by using inverse scattering transform technique, for instance .

Recently, a number of papers (such as ) focus on complex difference equations and difference analogues of Nevanlinna's theory. Ablowitz et al.  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 deg w R 2 .

Recently, Halburd and Korhonen  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 ) = π 1 z + π 2 w ( z ) + κ 1 , (4b) w ( z + 1 ) - w ( z ) + w ( z - 1 ) = π 1 z + ( - 1 ) z π 2 w ( z ) + κ 1 , (4c) w ( z + 1 ) + w ( z - 1 ) = π 1 z + π 3 w ( z ) + π 2 , (4d) w ( z + 1 ) + w ( z - 1 ) = π 1 z + κ 1 w ( z ) + π 2 w 2 ( z ) , (4e) w ( z + 1 ) + w ( z - 1 ) = π 1 z + κ 1 + π 2 ( - 1 ) z - w 2 ( z ) , (4f) w ( z + 1 ) + w ( z - 1 ) = π 1 z + κ 1 + π 2 1 - w 2 ( 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 π k , κ k 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  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 ) w 2 ( z ) - λ ( z ) w ( z ) + μ ( z ) ( w ( z ) - 1 ) ( w ( z ) - ν ( z ) ) , (7b) w ( z + 1 ) w ( z - 1 ) = η ( z ) w 2 ( 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 ) w m ( z ) . In (7a), the coefficients satisfy κ 2 ( z ) μ ( z + 1 ) μ ( z - 1 ) = μ 2 ( 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 ) ;

λ ( z + 1 ) λ ( z - 1 ) = λ ( z + 2 ) λ ( z - 2 ) , η ( z + 1 ) λ ( z + 1 ) = λ ( z + 2 ) η ( z - 1 ) , η ( z ) η ( z - 1 ) = η ( z + 2 ) η ( z - 3 ) ;

η ( z + 2 ) η ( z - 2 ) = η ( z ) η ( z - 1 ) , λ ( z ) = η ( z - 1 ) ;

λ ( z + 3 ) λ ( z - 3 ) = λ ( z + 2 ) λ ( z - 2 ) λ ( z ) , η ( z ) λ ( z ) = η ( z + 2 ) η ( z - 2 ) .

In (7d), h ( z ) 𝒮 ( w ) and m , | m | 2 .

Zhang and Yang  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 Δ w ( z ) = w ( z + 1 ) - w ( z ) . Then

w has no Nevanlinna exceptional value;

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

Example 2.

The function w ( z ) = z ( e i ( π / 2 ) z - 1 ) / ( e i ( 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) Δ w ( z ) = i e i π z + ( i - 1 ) ( 2 z + 1 ) e i ( π / 2 ) z - 1 ( i e i ( π / 2 ) z + 1 ) ( e i ( π / 2 ) z + 1 ) , Δ w ( z ) w ( z ) = i e i π z + ( i - 1 ) ( 2 z + 1 ) e i ( π / 2 ) z - 1 z ( i e i ( π / 2 ) z + 1 ) ( e i ( π / 2 ) z - 1 ) . Thus, (11) λ ( Δ w ) = λ ( 1 Δ w ) = σ ( w ) = 1 , λ ( Δ w w ) = λ ( 1 Δ w / 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 ( Δ w ) = λ ( 1 / Δ w ) = σ ( w ) , λ ( Δ w / w ) = λ ( 1 / ( Δ w / 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 , Δ w / w ) = T ( r , w ) + S ( r , w ) .

Example 5.

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

w has no Nevanlinna exceptional value;

λ ( Δ w ) = λ ( 1 / Δ w ) = σ ( w ) , λ ( Δ w / w ) = λ ( 1 / ( Δ w / 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 , Δ w / w ) = 2 T ( r , w ) + S ( r , w ) .

Example 9.

The function w ( z ) = z ( e i π z - 1 ) / ( e i π z + 1 ) is a meromorphic solution of difference equation (18) w ( z + 1 ) w ( z - 1 ) w 2 ( z ) = z 4 - z 2 , where h ( z ) = z 4 - z 2 . By calculation, this solution satisfies (19) Δ w ( z ) = e i 2 π z + ( 4 z + 2 ) e i π z + 1 e i 2 π z - 1 , Δ w ( z ) w ( z ) = e i 2 π z + ( 4 z + 2 ) e i π z + 1 z ( e i π z - 1 ) 2 . Thus, (20) λ ( Δ w ) = λ ( 1 Δ w ) = σ ( w ) = 1 , λ ( Δ w w ) = λ ( 1 Δ w / w ) = σ ( w ) = 1 .

From the following proofs of Theorems 17, 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 17 still hold.

Equations (7a)–(7c) and w ( z + 1 ) w ( z - 1 ) = h ( z ) w 2 ( 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 ( r σ - 1 + ε ) + O ( log r ) .

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 ( r λ - 1 + ε ) + O ( log r ) .

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 , 1 f ( z + c ) ) = N ( r , 1 f ( z ) ) + S ( r , f ) .

Lemma 15 (Valiron-Mohon'ko [<xref ref-type="bibr" rid="B12">18</xref>]).

Let f ( z ) be a meromorphic function. Then for all irreducible rational functions in f , (25) R ( z , f ( z ) ) = a n ( z ) f ( z ) n + + a 0 ( z ) b m ( z ) f ( z ) m + + b 0 ( z ) with meromorphic coefficients a i ( z ) , b j ( 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 , 1 w - 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 deg f U ( z , f ) = n in f ( z ) and its shifts and deg f Q ( 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 ( r σ - 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 , we know P ( z , a ) = a 2 - 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 { } , δ ( a , w ) = 0 . So, w has no Nevanlinna exceptional value.

(ii) First, we prove that λ ( Δ w / w ) = λ ( 1 / ( Δ w / w ) ) = σ ( w ) . By (8) and Lemma 12, we obtain (33) 2 T ( r , w ( z ) ) = T ( r , h ( z ) w 2 ( z ) ) + O ( log r ) = T ( r , w ( z + 1 ) w ( z ) w ( z - 1 ) w ( z ) ) + O ( log r ) T ( r , w ( z + 1 ) w ( z ) ) + T ( r , w ( z ) w ( z - 1 ) ) + O ( log r ) = 2 T ( r , w ( z + 1 ) w ( z ) ) + S ( r , w ( z + 1 ) w ( z ) ) + O ( log r ) 2 T ( 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 , Δ w ( z ) w ( z ) ) + S ( r , w ( z ) ) . From (34) and Lemmas 11 and 12, we deduce that (35) N ( r , Δ w w ) = T ( r , Δ w w ) - m ( r , Δ w w ) = T ( r , Δ w w ) + S ( r , w ) T ( r , w ) + S ( r , w ) . Thus, λ ( 1 / ( Δ w / w ) ) σ ( w ) , that is, λ ( 1 / ( Δ w / 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 P 1 ( 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 P 1 ( z , 1 ) = 1 - ( h ( z + 1 ) / h ( z ) ) 0 . Since P 1 ( z , 1 ) 0 , by (37) and Lemmas 12 and 16, we have (38) m ( r , 1 g ( z ) - 1 ) = S ( r , g ( z ) ) = S ( r , w ( z + 1 ) w ( z ) ) S ( r , w ( z ) ) . Thus, (39) m ( r , 1 Δ w ( z ) / w ( z ) ) = m ( r , 1 ( w ( z + 1 ) / w ( z ) ) - 1 ) = m ( r , 1 g ( z ) - 1 ) = S ( r , w ( z ) ) . By (34) and (39), we have (40) N ( r , 1 Δ w / w ) = T ( r , Δ w w ) - m ( r , 1 Δ w / w ) = T ( r , Δ w w ) + S ( r , w ) T ( r , w ) + S ( r , w ) . Then, λ ( Δ w / w ) σ ( w ) , that is, λ ( Δ w / w ) = σ ( w ) .

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

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

Set a = 0 in (i). By (39), we obtain (46) m ( r , 1 Δ w ) = m ( r , 1 Δ w / w 1 w ) m ( r , 1 Δ w / w ) + m ( r , 1 w ) = S ( r , w ) . Combining this with (43), we have (47) N ( r , 1 Δ w ) = T ( r , Δ w ) - m ( r , 1 Δ w ) = T ( r , Δ w ) + S ( r , w ) T ( r , w ) + S ( r , w ) . Then λ ( Δ w ) σ ( w ) , that is, λ ( Δ w ) = 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 ) w 2 ( z ) w ( z + 1 ) w ( z - 1 ) ) = S ( r , w ) , m ( r , 1 w ( z ) ) = m ( r , 1 w ( z ) w ( z + 1 ) w ( z - 1 ) h ( z ) w ( z ) ) = m ( r , w ( z + 1 ) w ( z - 1 ) h ( z ) w 2 ( z ) ) = S ( r , w ) . Hence, (49) N ( r , w ( z ) ) = T ( r , w ( z ) ) + S ( r , w ) , (50) N ( r , 1 w ( 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 { 0 } , we have P ( z , a ) = a 2 - a 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 . Combining with δ ( 0 , w ) = δ ( , w ) = 0 , we know w has no Nevanlinna exceptional value.

(ii) First, we prove λ ( Δ w ) = λ ( 1 / Δ w ) = σ ( w ) . Since ( z + 1 ) = w ( z ) + Δ w ( z ) ,  w ( z - 1 ) = w ( z ) - Δ w ( z - 1 ) , by (12), we have (52) ( w ( z ) + Δ w ( z ) ) ( w ( z ) - Δ w ( z - 1 ) ) = h ( z ) w ( z ) ; that is, (53) ( Δ w ( z ) - Δ w ( z - 1 ) ) w ( z ) - Δ w ( z ) Δ w ( z - 1 ) = - w 2 ( z ) + h ( z ) w ( z ) .

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

If z 1 is a pole of w ( z ) with multiplicity k , not pole of h ( z ) , then z 1 is a pole of - w 2 ( z ) + h ( z ) w ( z ) with multiplicity 2 k . From (53), one of Δ w ( z ) and Δ w ( z - 1 ) must have the pole z 1 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 , Δ w ( z ) ) + N ( r , Δ w ( z - 1 ) ) + N ( r , h ( z ) ) + S ( r , w ) = 2 N ( r , Δ w ( z ) ) + S ( r , Δ w ( z ) ) + O ( log r ) + S ( r , w ) 2 N ( r , Δ w ( z ) ) + S ( r , w ) . Hence, σ ( w ) λ ( 1 / Δ w ) , that is, λ ( 1 / Δ w ) = σ ( w ) .

Next, we prove that λ ( Δ w / w ) = λ ( 1 / ( Δ w / w ) ) = σ ( w ) . By (12), we have (56) Δ w ( 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 , Δ w ( z ) w ( z ) ) = T ( r , Δ w ( z ) w ( z ) ) - m ( r , Δ w ( z ) w ( z ) ) = T ( r , Δ w ( 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 / ( Δ w / 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 , 1 w ( z ) - h ( z + 1 ) ) = T ( r , w ( z ) ) + S ( r , w ( z ) ) . By (56), if z 0 is a common zero of h ( z ) - w ( z - 1 ) and w ( z - 1 ) , then z 0 must be a zero of h ( z ) . Thus, by (56), (58), and Lemma 14, we have (59) N ( r , 1 Δ w ( z ) / w ( z )    ) N ( r , 1 w ( z - 1 ) - h ( z ) ) - N ( r , 1 h ( z ) ) = N ( r , 1 w ( z ) - h ( z + 1 ) ) + O ( log r ) + S ( r , w ( z ) ) = T ( r , w ( z ) ) + S ( r , w ( z ) ) . Hence, λ ( Δ w / w ) σ ( w ) , that is, λ ( Δ w / 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 , we have P ( z , a ) = a 3 - 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 { } , δ ( a , w ) = 0 . So, w has no Nevanlinna exceptional value.

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

By (62), we obtain (64) T ( r , Δ w ) = T ( r , Δ w w w ) T ( r , Δ w w ) - T ( r , w ) 1 2 T ( r , w ) + S ( r , w ) . By (60), (64), and Lemma 11, we have (65) N ( r , Δ w ) = T ( r , Δ w ) - m ( r , Δ w ) T ( r , Δ w ) - m ( r , Δ w w ) - m ( r , w ) = T ( r , Δ w ) + S ( r , w ) 1 2 T ( r , w ) + S ( r , w ) . Then λ ( 1 / Δ w ) σ ( w ) , that is, λ ( 1 / Δ w ) = σ ( w ) .

Next, we prove that λ ( Δ w ) = λ ( Δ w / 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 P 1 ( 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, P 1 ( z , 1 ) = 1 - ( h ( z + 1 ) / h ( z ) ) 0 . By this and by (68) and Lemmas 12 and 16, we obtain (69) m ( r , 1 g ( z ) - 1 ) = S ( r , g ( z ) ) = S ( r , w ( z + 1 ) w ( z ) ) S ( r , w ( z ) ) . That is, (70) m ( r , 1 Δ w ( z ) / w ( z ) ) = m ( r , 1 ( w ( z + 1 ) / w ( z ) ) - 1 ) = m ( r , 1 g ( z ) - 1 ) = S ( r , w ( z ) ) . By (62) and (70), we have (71) N ( r , 1 Δ w / w ) = T ( r , Δ w w ) - m ( r , 1 Δ w / w ) + O ( 1 ) = T ( r , Δ w w ) + S ( r , w ) 3 2 T ( r , w ) + S ( r , w ) . Thus, λ ( Δ w / w ) σ ( w ) , that is, λ ( Δ w / w ) = σ ( w ) .

Set a = 0 in (i). By (70), we have (72) m ( r , 1 Δ w ) = m ( r , 1 Δ w / w 1 w ) m ( r , 1 Δ w / w ) + m ( r , 1 w ) = S ( r , w ) . Thus, by (64), (73) N ( r , 1 Δ w ) = T ( r , Δ w ) - m ( r , 1 Δ w ) = T ( r , Δ w ) + S ( r , w ) 1 2 T ( r , w ) + S ( r , w ) . Hence, λ ( Δ w ) σ ( w ) ) , that is, λ ( Δ w ) = 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) 4 T ( r , w ( z ) ) = T ( r , h ( z ) w 4 ( z ) ) + O ( log r ) = T ( r , w ( z + 1 ) w ( z ) w ( z - 1 ) w ( z ) ) + O ( log r ) T ( r , w ( z + 1 ) w ( z ) ) + T ( r , w ( z ) w ( z - 1 ) ) + O ( log r ) = 2 T ( r , w ( z + 1 ) w ( z ) ) + S ( r , w ( z + 1 ) w ( z ) ) + O ( log r ) 4 T ( r , w ( z ) ) + S ( r , w ( z ) ) . Thus, (75) T ( r , Δ w ( z ) w ( z ) ) = T ( r , w ( z + 1 ) w ( z ) ) + O ( 1 ) = 2 T ( r , w ( z ) ) + S ( r , w ( z ) ) . By (75) and Lemma 11, we know (76) N ( r , Δ w w ) = T ( r , Δ w w ) - m ( r , Δ w w ) = T ( r , Δ w w ) + S ( r , w ) = 2 T ( r , w ) + S ( r , w ) . Therefore, λ ( 1 / ( Δ w / w ) ) = σ ( w ) .

By (17), we know (77) w ( z + 2 ) w ( z ) w 2 ( 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 ) g 2 ( z ) = h ( z + 1 ) / h ( z ) . Set P ( z , g ) : = g ( z + 1 ) g ( z - 1 ) g 2 ( 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 , 1 g ( z ) - 1 ) = S ( r , g ( z ) ) = S ( r , w ( z + 1 ) w ( z ) ) S ( r , w ( z ) ) , thus, (81) m ( r , 1 Δ w ( z ) / w ( z ) ) = m ( r , 1 ( w ( z + 1 ) / w ( z ) ) - 1 ) = m ( r , 1 g ( z ) - 1 ) = S ( r , w ( z ) ) . By this and (75), we have (82) N ( r , 1 Δ w / w ) = T ( r , Δ w w ) - m ( r , 1 Δ w / w ) = T ( r , Δ w w ) + S ( r , w ) = 2 T ( r , w ) + S ( r , w ) . Then λ ( Δ w / w ) = σ ( w ) .

We see from (76) that (83) N ( r , Δ w ) = N ( r , Δ w w w ) N ( r , Δ w w ) - N ( r , 1 w ) N ( r , Δ w w ) - T ( r , w ) = T ( r , w ) + S ( r , w ) .

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

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