Poles and α-points of Meromorphic Solutions of the First Painlevé Hierarchy

Transcription

1 Publ. RIMS, Kyoto Univ ), Poles and α-points of Meromorphic Solutions of the First Painlevé Hierarchy By Shun Shimomura Abstract The first Painlevé hierarchy, which is a sequence of higher order analogues of the first Painlevé equation, follows from the singular manifold equations for the mkdv hierarchy. For meromorphic solutions of the first Painlevé hierarchy, we give a lower estimate for the number of poles; which is regarded as an extension of one corresponding to the first Painlevé equation, and which indicates a conjecture on the growth order. From our main result, two corollaries follow: one is the transcendency of meromorphic solutions, and the other is a lower estimate for the frequency of α- points. An essential part of our proof is estimation of certain sums concerning the poles of each meromorphic solution. 1. Introduction For a meromorphic function fz) in the whole complex plane C, the counting function for poles of fz) is defined by r Nr, f):= nρ, f) n0,f)) dρ ρ + n0,f)logr, 0 where nr, f) denotes the number of poles inside the disk z r, each counted according to its multiplicity. Moreover, we use the notation cf. [6], [8]): 2π mr, f):= 1 log + fre iφ ) dφ, 2π 0 T r, f):=mr, f)+nr, f) log + x := max{0, log x}, Communicated by T. Kawai. Received May 15, Mathematics Subject Classifications): 34M55, 34M05, 30D35 Department of Mathematics, Keio University, , Hiyoshi, Kohoku-ku, Yokohama , Japan. shimomur@math.keio.ac.jp

2 472 Shun Shimomura denoting, respectively, the proximity and the characteristic functions; the growth order of fz) is defined by σf) := lim sup r log T r, f). log r Let wz) be an arbitrary solution of the first Painlevé equation PI) w =6w 2 + z = d/dz). Then, wz) is a transcendental meromorphic function. By a wellknown argument in the Nevanlinna theory, lim sup r Nr, w)/ log r) = cf. Remark 1.2), which implies that wz) admits infinitely many poles. This fact is quantitatively represented as 1.1) lim sup r log Nr, w) log r 5 2 cf. [11]); and, more precisely, 1.2) Nr, w) r5/2 log r cf. [5, 7], [13]). These combined with T r, w) r 5/2 cf. [5, 8], [12], [14]) imply that the growth order of wz) is equal to 5/2. For real-valued functions φr) andψr) ontheintervalr 0, + ), we write φr) ψr) orψr) φr) if φr) =Oψr)) as r +. In the case where hz) is a function of z C, we also write hz) ψr) if hz) = Oψr)) as z = r +.) A sequence of higher order analogues of PI) is given in the following manner cf. [5, 16], [7]). Let d ν [w] ν =0, 1, 2,...) be differential polynomials in w determined by the recursion relation 1.3) 1.4) d 0 [w]=1, Dd ν+1 [w]=d 3 8wD 4w )d ν [w], D = d/dz, ν N {0} cf. Lemma 2.6 with its proof). Some of them are written in the form d 1 [w]/4= w + C 10, d 2 [w]/4= w +6w 2 + C 21 d 1 [w]+c 20, d 3 [w]/4= w 4) +20ww + 10w ) 2 40w 3 + C 32 d 2 [w]+c 31 d 1 [w]+c 30, d 4 [w]/4= w 6) +28ww 4) +56w w 3) + 42w ) 2 280w 2 w + ww ) 2 w 4 ) + C 43 d 3 [w]+c 42 d 2 [w]+c 41 d 1 [w]+c 40,

3 First Painlevé Hierarchy 473 where C ij are arbitrary constants. Consider a sequence of 2ν-th order equations of the form PI 2ν ) d ν+1 [w]+4z =0, ν N, which is called the first Painlevé hierarchy. EquationPI 2 ) essentially coincides with PI). These equations follow from the singular manifold equations for the mkdv hierarchy cf. [7], [9], [15]). As in the case of PI), it is basic and interesting to study analytic properties of meromorphic solutions of PI 2ν ), for example, to determine the growth order of them. The purpose of this paper is to show the following, which is an extension of 1.1), and which is a first step toward this question: Theorem 1.1. Suppose that w ν z) be a meromorphic solution of PI 2ν ). Then we have 1.5) lim sup r log Nr, w ν ) log r 2ν +3 ν +1, namely the growth order of w ν z) is not less than 2ν +3)/ν +1). Corollary 1.2. Equation PI 2ν ) admits no rational solutions. Furthermore, the frequency of α-points is estimated as follows: Corollary 1.3. For each α C, 1.6) log Nr, 1/w ν α)) 2ν +3 lim sup r log r ν +1. Theorem 1.1 with the special case ν = 1 leads us to the following: Conjecture. The growth order of w ν z) is equal to 2ν +3)/ν +1). Remark 1.1. Since Nr, w ν ) nr, w ν )logr, the quantity Nr, w ν )in 1.5) can be replaced by nr, w ν ). Remark 1.2. For a meromorphic function fz), the deficiency of is defined by δ,f) := lim inf r mr, f)/t r, f)) [6], [8]). For every solution wz) ofpi),wehaveδ,w) = 0; and this fact combined with the transcendency of wz) implies lim sup r Nr, w)/ log r) = cf. e.g. [5, 10]). For w ν z), analogous deficiency relations are valid: δ,w ν )=0, and for each α C, δα, w ν ):=δ, 1/w ν α)) = 0 cf. 3.7) and 4.1) of the proofs of Theorem 1.1 and Corollary 1.3).

4 474 Shun Shimomura Remark 1.3. The second Painlevé equation PII) w =2w 3 + zw + a, a C belongs to the second Painlevé hierarchy PII 2ν )ν N) cf. [1], [2], [3], [7]). Value distribution properties of solutions of PII 2ν ) are studied by Gromak and He [4]) and by Li and He [10]); for example, every transcendental meromorphic solution w II,ν z) satisfiesδ,w II,ν )=0. Theorem 1.1 and its corollaries are proved in Sections 3 and 4. In the proofs, we need some basic facts in the Nevanlinna theory and some properties of differential polynomials. They are reviewed or explained in Section 2. To prove Theorem 1.1, we deal with certain sums concerning the poles of each meromorphic solution, which are essential in the proof; and these sums are evaluated in the final section. 2. Basic Facts 2.1. Nevanlinna Theory We review basic facts in the Nevanlinna theory which are necessary in the proofs of our results cf. [5, Appendix B], [6], [8, Chapters 1 and 2]). Let fz) be an arbitrary non-constant meromorphic function. Lemma 2.1. For an arbitrary α C, T r, 1/f α)) = T r, f)+o1). Lemma 2.2. i) T r, f) is a monotone increasing function of r. ii) T r, f) log r if and only if fz) is a rational function. iii) If fz) is transcendental, then T r, f)/ log r as r. The following lemmas [5, Lemmas B.11 and B.12], [8, Lemma and Proposition 9.2.3]) are useful in the study of differential equations. Lemma 2.3. Let fz) be a non-constant meromorphic function satisfying f λ+1 = P z, f) λ N), where P z, u) is a polynomial in z, u and derivatives of u. Suppose that the total degree of P z, u) with respect to u and its derivatives does not exceed λ. Thenmr, f) log T r, f)+logr as r outside an exceptional set with finite linear measure. Lemma 2.4. Let fz) be a non-constant meromorphic function satisfying F z, f) =0, where F z, u) is a polynomial in z, u and derivatives of u. Suppose that α C satisfies F z, α) 0. Then mr, 1/f α)) log T r, f)+logr as r outside an exceptional set with finite linear measure.

5 First Painlevé Hierarchy 475 For real-valued functions, we have [5, Lemma B.10], [8, Lemma 1,1,1]) Lemma 2.5. Let φr) and ψr) be real-valued monotone increasing functions on 0, + ). Suppose that φr) ψr) outside an exceptional set with finite linear measure. Then φr) ψ2r) on r 0, + ), where r 0 is some positive number Differential Polynomials For an arbitrary nonnegative integer p N {0}, set p [w, w,...,w p) ] ι := w κ) ) ι κ, w 0) = w, κ=0 where ι =ι 0,ι 1,...,ι p ) N {0}) p+1. For the index ι, we put p ι := κ +2)ι κ. Consider the differential polynomial κ=0 ϕ[w] = ι I ϕ c ι [w, w,...,w p) ] ι, c ι C\{0}, where I ϕ N {0}) p+1 is a finite set of indices. For ϕ[w] 0), we define its weight by wtϕ[w]) = max{ ι ι I ϕ }; in particular, if ϕ[w] c 0 C\{0}, then wtc 0 )=0. For any integer q>p, the differential polynomial ϕ[w] admits another expression ϕ[w] = ι I ϕ c ι [w, w,...,w p),...,w q) ] ι,o) with ι, o) =ι 0,ι 1,...,ι p, 0,...,0). Then, wtϕ[w]) = wt ϕ[w]), namely the definition of the weight is independent of the choice of the size q of the index. Let ϕ[w] 0) and ψ[w] 0) be arbitrary differential polynomials. Then, wtϕ[w]+ψ[w]) = max{wtϕ[w]), wtψ[w])}, wtϕ[w]ψ[w]) = wtϕ[w]) + wtψ[w]).

6 476 Shun Shimomura Remark 2.1. As will be shown later cf. Lemma 2.7), every pole of w ν z) is double. Hence, for every pole of [w ν,w ν,...,w ν 2ν) ] ι, the multiplicity of it is equal to ι. This fact is a background of the definition of the weight of differential polynomials. 2.1) where We note the following: Lemma 2.6. For each ν N {0}, d ν+1 [w] is expressible in the form d ν+1 [w] =γ ν+1 w ν+1 + ι 2ν+1) ι 0 ν i) ι =ι 0,ι 1,...,ι 2ν ) N {0}) 2ν+1, ii) γ ν+1 C \{0}, c ι C. c ι [w, w,...,w 2ν) ] ι, Proof. By 1.4), for every ν N, ν d ν µ [w]dd µ+1 [w] = and hence, by 1.3), ν d ν µ [w]d 3 8wD 4w )d µ [w], ν 1 Dd ν+1 [w] = d ν µ [w]dd µ+1 [w] + ν dν µ [w]d 3 8wd ν µ [w]d 4w d ν µ [w] ) d µ [w]. Substituting the identities ν 1 d ν µ [w]dd µ+1 [w] = 1 ν 1 ) 2 D d ν µ [w]d µ+1 [w], ν ν ) d ν µ [w]d 3 d µ [w] =D d ν µ [w]d 2 d µ [w] 1 ν Dd ν µ [w] Dd µ [w], 2 ) ν w d ν µ [w]dd µ [w] = 1 ν ν w 2 D d ν µ [w]d µ [w] w d ν µ [w]d µ [w], 2

7 First Painlevé Hierarchy 477 we have ν ) Dd ν+1 [w]=d d ν µ [w]d 2 1 ) 2 Dd ν µ[w] D 4wd ν µ [w] d µ [w] which implies that 2.2) d ν+1 [w]= 1 ν 1 ) 2 D d ν µ [w]d µ+1 [w], ν d ν µ [w]d 2 1 ) 2 Dd ν µ[w] D 4wd ν µ [w] d µ [w] 1 ν 1 d ν µ [w]d µ+1 [w]+c ν, 2 where C ν is an arbitrary constant. By 2.2) combined with 1.3), d ν+1 [w] isa differential polynomial. Moreover, in d ν+1 [w], the derivative with the highest order is w 2ν). Indeed, this fact is inductively checked by using 1.3) and 1.4). Hence, for every ν N {0}, d ν+1 [w] is written in the form 2.3) d ν+1 [w] = ι I ν c ι [w, w,...,w 2ν) ] ι, ι =ι 0,ι 1,...,ι 2ν ), c ι C\{0}, where I ν N {0}) 2ν+1 is a finite set of indices. We prove 2.1) by induction on ν. Clearly it is valid for ν =0. Suppose that 2.1) is valid for every ν N; namely, wtd ν+1 [w]) 2ν + 1) for every ν N, and γ N+1 C\{0}. Since, for ι o, ) 2ν wtd[w, w,...,w 2ν) ] ι )) = wt [w, w,...,w 2ν) ] ι ι µ w µ+1) /w µ) =max { wtw ι 0 w ) ι1 w 2ν) ) ι 2ν w µ+1) /w µ) ) 0 µ 2ν, ι µ 0 } =max { ι +µ +2) 1) + µ +3) 1 0 µ 2ν, ιµ 0 } = ι +1 = wt[w, w,...,w 2ν) ] ι )+1, we have wtd l d ν+1 [w]) = l +wtd ν+1 [w]) l +2ν +1) for ν N and for l =1, 2. Hence, by 2.2) with ν = N +1, wtd N+2 [w]) 2N +1)+2 = 2N +2). Furthermore, by 1.4), Dd N+2 [w]= 8wDγ N+1 w N+1 ) 4w γ N+1 w N+1 + = 4γ N+1 2N +3)w N+1 w +,

8 478 Shun Shimomura which implies that γ N+2 = 42N +3)N +2) 1 γ N+1 C\{0}. Hence, 2.1) is valid for ν = N + 1 as well. This completes the proof. Lemma 2.7. For a meromorphic solution w ν z) of PI 2ν ), let a 0 be an arbitrary pole of it. Then, around z = a 0, w ν z) =ca 0 )z a 0 ) 2 + O1), where ca 0 )=ka 0 )ka 0 )+1)/2 for some integer ka 0 ) {1,...,ν}. Proof. Around the pole z = a 0, we write w ν z) =bz a 0 ) σ +, b 0. Suppose that σ 3. It is inductively shown that d k [w ν ]z) = b k z a 0 ) σk +,b k 0foreveryk N, because this formula with k implies namely Dd k+1 [w ν ]z)=d 3 8w ν z)d 4w νz))d k [w ν ]z) = 8 σk) 4 σ))bb k z a 0 ) σk+1) 1 + =42k +1)σbb k z a 0 ) σk+1) 1 +, d k+1 [w ν ]z) = 42k +1)k +1) 1 bb k z a 0 ) σk+1) +. Hence, if σ 3, substitution of d ν+1 [w ν ]z) =b ν+1 z a 0 ) σν+1) + into PI 2ν ) yields a contradiction. Supposing that σ =1, by an analogous argument, we can show that d ν+1 [w ν ]z) =b ν+1z a 0 ) 2ν+1) +,b ν+1 0, and also derive a contradiction. Therefore, z = a 0 is a double pole. Put w ν z) = b 0 z a 0 ) 2 +,b 0 0. Since substitution of d k [w ν ]z) =A k z a 0 ) 2k +, k N into 1.4) yields that d k+1 [w ν ]z) =A k+1 z a 0 ) 2k+1) + with By this fact, A k+1 = 42k +1)k +1) 1 b 0 kk +1)/2)A k. A ν+1 = B ν+1 b 0 ν b 0 kk +1)/2), B ν+1 0. k=1 Substituting d ν+1 [w ν ]z) intopi 2ν ), we have b 0 = kk +1)/2 forsomek {1,...,ν}. Furthermore, the relation Dd ν+2 [w ν ]z)=d 3 8w ν z)d 4w νz))d ν+1 [w ν ]z) =D 3 8w ν z)d 4w νz)) 4z) =32w ν z)+16zw νz) =16Dzw ν z)) + 16w ν z),

9 First Painlevé Hierarchy 479 namely 16w ν z) =Dd ν+2 [w ν ]z) 16zw ν z)) means that the residue of w ν z) atthepolez = a 0 vanishes. This completes the proof. 3. Proof of Theorem 1.1 To prove 1.5), we suppose the contrary: 3.1) lim sup r log Nr, w ν ) log r < 2ν +3 ν +1, namely, for some ε>0, Nr, w ν ) r 2ν+3)/ν+1) ε, from which it follows that 3.2) nr) =nr, w ν ) r 2ν+3)/ν+1) ε, because N2r, w ν ) 2r r nρ, wν ) n0,w ν ) ) dρ ρ nr, w ν )+O1) ) log 2. Starting from 3.1), we would like to derive a contradiction. Let {a j } j J be a sequence of all distinct poles of w ν z) arranged as a 1 a j, where J = N or {1,...,p} p N) or. Clearly these poles do not accumulate at any point in C. By Lemma 2.7, we write w ν z) intheform 3.3) 3.4) w ν z) =Φz)+gz), Φz) = j J ca j ) z a j ) 2 a 2 j ), where gz) is an entire function. In 3.4), we make the following conventions: i) if a 1 =0, then the term z a 1 ) 2 a 2 1 is replaced by z 2 ; ii) if J =, then Φz) 0. In what follows, we may suppose that Φz) 0, because the case where Φz) 0 is similarly treated by adding a slight modification. Under 3.2), we have the following lemmas whose proofs will be given afterward: Lemma 3.1. For every r>1, there exists z r satisfying 0.7r z r r, z r a j 2 r 1/ν+1) ε/2, z r a j 3 r 3/2)/ν+1) ε.

10 480 Shun Shimomura Lemma 3.2. Let r be an arbitrary number satisfying r>1. Then, z aj ) 2 a 2 r 1/ν+1) ε, z a j 3 1 for z r, and Lemma 3.3. such that 0< a j < j 0< a 2 j r 1/ν+1) ε. There exists a set E 0, ) with finite linear measure z aj ) 2 a 2 z 9 j for z 0, ) \ E. By Lemma 2.6, w ν z) satisfies the equation γ ν+1 w ν+1 = c ι w ι 0 w ) ι1 w 2ν) ) ι 2ν 3.5) +4z. ι 2ν+1) ι 0 ν For each term on the right-hand side, note that 3.6) 2ν κ=0 ι κ ν, because 2 2ν κ=0 ι κ = ι 2ν κ=0 κι κ =2ν + 1) is valid if and only if ι = ι 0,ι 1,...,ι 2ν )=ν+1, 0,...,0). By Lemma 2.3, there exists a set E 0, ) with finite linear measure such that 3.7) mr, w ν ) log T r, w ν )+logr as r,r E. Note that PI 2ν ) does not admit a constant solution.) Combining this with 3.1), we have T r, w ν ) r 2ν+3)/ν+1) and mr, w ν ) log r for r E. By Lemma 3.3, for r E E, T r, g) =mr, g) =mr, w ν Φ) mr, w ν )+mr, Φ) log r. By Lemmas 2.5 and 2.2, this is valid for r approaching without an exceptional set, and hence gz) isapolynomial. By Lemmas 3.1 and 3.2, for every r>1, there exists z r, 0.7r z r r satisfying 3.8) Φz r ) + z r a j 2 + 0< a 2 j z r a j ) 2 a 2 j r 1/ν+1) ε/2.

11 First Painlevé Hierarchy 481 Then, also for every κ =1, 2,...,2ν, 3.9) Φ κ) z r ) r κ/2+1)/ν+1) ε. Indeed, observing that Φ κ) z r ) we have the following: i) if κ is odd, ii) if κ is even, Φ κ) z r ) Φ κ) z r ) From 3.5), we have 3.10) w ν z r ) z r + + z r a j 2 κ + z r a j 3 ) z r a j 3 r 3/2)/ν+1) ε r 1/ν+1) ε/2)κ 1)/2 r κ/2+1)/ν+1) ε ; z r a j 2 ) κ/2+1 z r a j 2 κ, z r a j 2 ) κ 1)/2 + r 1/ν+1) ε/2)κ/2+1) r κ/2+1)/ν+1) ε. ι 2ν+1) ι 0 ν z r a j 3 1/ν+1) w ν z r ) ι 0 w νz r ) ι1 w ν 2ν) z r ) ι 2ν ). Now suppose that deg gz) =δ 0 1. Substitute w κ) ν z r )=g κ) z r )+ Φ κ) z r )κ =0, 1,...,2ν) into 3.10), and observe that w ν z r ) gz r ) Φz r ) = gz r ) + Or 1/ν+1) ) r δ 0, and that, for every ι satisfying ι 0 ν and ι 2ν +1), w ν z r ) ι 0 w νz r ) ι1 w ν 2ν) z r ) ι 2ν gz r ) + Φz r ) ) ι 0 gz r ) + Φ z r ) ) ι1 gz r ) + Φ 2ν) z r ) ) ι 2ν z r δ 0ι 0 +ι 1 + +ι 2ν ) r δ 0ν

12 482 Shun Shimomura cf. 3.6), 3.8) and 3.9)). Then, we have the contradiction r δ 0 z r + r δ0ν ) 1/ν+1) r δ0ν/ν+1) ; which implies that gz) C C. Substituting w = w ν z) =Φz) +C into PI 2ν ), and observing that, for every ι satisfying 0 < ι 2ν +1), w ν z r ) ι 0 w νz r ) ι1 w ν 2ν) z r ) ι 2ν Φz r ) + C ) ι 0 Φ z r ) ι1 Φ 2ν) z r ) ι 2ν r χι) ε/2 with χι) = 2ν κ=0 κ/2 +1)ι κ/ν +1) = ι /2)/ν +1) 1 cf. 3.8) and 3.9)), we have 0.7r z r d ν+1 [w ν ]z r ) r 1 ε/2, which is a contradiction. We have thus proved 1.5). 4. Proofs of Corollaries 1.2 and 1.3 Corollary 1.2 immediately follows from Theorem 1.1. To prove Corollary 1.3, note that w α C) is not a solution of PI 2ν ). By Lemma 2.4, 4.1) mr, 1/w ν α)) log T r, w ν )+logr as r for r E 1, where E 1 0, ) is a set with finite linear measure. Since w ν z) is transcendental, by Lemmas 2.1 and 2.2, we have Nr, 1/w ν α)) T r, w ν ) and hence =1 mr, 1/w ν α)) + O1) T r, w ν ) 1 as r, r E 1 ; 4.2) Nr, 1/w ν α)) 1/2)T r, w ν ) for r r 1, ) \ E 1, for some r 1 > 0. On the other hand, by 3.7), Nr, w ν )/T r, w ν ) 1asr, r E. Hence, 4.3) 1/2)Nr, w ν ) T r, w ν ) for r r 2, ) \ E, for some r 2 > 0. Using Lemma 2.5, from 4.2) and 4.3), we derive that Nr, w ν ) 4N2r, 1/w ν α)) for r r 3, ), where r 3 > 0 is sufficiently large. This inequality combined with 1.5) yields the conclusion 1.6) of Corollary 1.3.

13 First Painlevé Hierarchy Proofs of Lemmas 3.1, 3.2 and Proof of Lemma 3.1 Put D r = {z z r} and 0 = C \ j 1 U j) ;whereuj = {z z a j < a j 1/2)/ν+1) } if a j 0, and U 1 = {z z < 1} if a 1 =0. Since, by 3.2), r a j 1/ν+1) = ρ 1/ν+1) dnρ) 1< a j <r = 1 [ ρ 1/ν+1) nρ) ] r ν +1 r 1 ρ 1 1/ν+1) nρ)dρ r 2 ε, we can take r 0 so large that 7πr 2 /8 µ 0 D r ) <πr 2 for every r>r 0, where µx) denotes the area of a set X. For every r>1, if a j < 2r, then z a j 2 dxdy ρ 1 dρdθ log r, D r \U j and Hence, 5.1) 5.2) 0 D r 0 D r D r \U j z a j 3 dxdy a j 1/2)/ν+1) ρ 3r 0 θ 2π a j 1/2)/ν+1) ρ 3r 0 θ 2π ρ 2 dρdθ r 1/2)/ν+1). z a j 2 dxdy n2r)logr K 0 r 2ν+3)/ν+1) ε/2, z a j 3 dxdy n2r)r 1/2)/ν+1) K 0 r 2+3/2)/ν+1) ε, where K 0 is some positive number. Consider the sets { Fr 1 = z 0 D r z a j 2 8π 1 K 0 r }, 1/ν+1) ε/2 { Fr 2 = z 0 D r z a j 3 8π 1 K 0 r 3/2)/ν+1) ε }. Suppose that µfr 1 ) < 3πr 2 /4. Then z a j 2 dxdy > 8π 1 K 0 r 1/ν+1) ε/2 7/8 3/4)πr 2 0 D r \F 1 r = K 0 r 2ν+3)/ν+1) ε/2,

14 484 Shun Shimomura which contradicts 5.1). This implies that µf 1 r ) 3πr 2 /4. By the same argument, we have µf 2 r ) 3πr 2 /4. Hence, µf 1 r F 2 r ) πr 2 /2. Observing that µ{z z < 0.7r}) =0.49πr 2, we have {z 0.7r z r} F 1 r F 2 r ), which implies the conclusion Proof of Lemma 3.2 For a j 2r, and for z D r, observing that z/a j 1/2, we have z a j 3 8 a j 3,and z a j ) 2 a 2 j =2 z a j 3 1 z/a j )/2 1 z/a j 2 10r a j 3. Hence, by 3.2), z aj ) 2 a 2 r and 0< z a j 3 a 2 j 2r Thus the lemma is proved. 1 j r 2r a j 3 t 2 dnt)+o1) r 1/ν+1) ε + We put E =0, a 1 +1) Since, by 3.2), a j 3 j J\{1} 2r 1 a j 3 r 2r t 3 dnt) t 4 nt)dt r 1/ν+1) ε, 2r 5.3. Proof of Lemma j J\{1} t 3 dnt)+o1) t 3 dnt) 1, t 3 nt)dt + O1) r 1/ν+1) ε. aj a j 3, a j + a j 3)). 1 t 4 nt)dt + O1) 1,

15 First Painlevé Hierarchy 485 the total length of E is finite. If z E, then + ) z aj ) 2 a 2 0< a j <2 z a j 2 z j z 6 +1)n2 z )+ z 1/ν+1) z 9. This completes the proof. References [1] Airault, H., Rational solutions of Painlevé equations,stud. Appl. Math., ), [2] Flaschka, H. and Newell, A. C., Monodromy- and spectrum-preserving deformations, I, Commun. Math. Phys., ), [3] Gromak, V. I., Nonlinear evolution equations and equations of P -type, Differential Equations, ), [4] Gromak, V. I. and He, Y., On the solutions of the second Painlevé equation of higher order, Proc. Math. Inst. Belarus Natl. Acad. Sci., ), [5] Gromak, V. I., Laine, I. and Shimomura, S., Painlevé Differential Equations in the Complex Plane, Walter de Gruyter, Berlin, New York, [6] Hayman, W. K., Meromorphic Functions, Clarendon Press, Oxford, [7] Kudryashov, N. A., The first and second Painlevé equations of higher order and some relations between them, Phys. Lett. A, ), [8] Laine, I., Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, New York, [9] Lax, P. D., Almost periodic solutions of the KdV equation, SIAM Rev., ), [10] Li, Y. and He, Y., On analytic properties of higher analogs of the second Painlevé equation, J. Math. Phys., ), [11] Mues, E. and Redheffer, R., On the growth of logarithmic derivatives, J. London Math. Soc., ), [12] Shimomura, S., Growth of the first, the second and the fourth Painlevé transcendents, Math. Proc. Cambridge Philos. Soc., ), [13], Lower estimates for the growth of Painlevé transcendents, Funkcial. Ekvac., ), [14] Steinmetz, N., Value distribution of Painlevé transcendents, Israel J. Math., ), [15] Weiss, J., On classes of integrable systems and the Painlevé property, J. Math. Phys., ),