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: Bibliography : Lower Estimates for the : Lower Estimates for the


Consider the first and the second Painlevé equations

(I)   $\displaystyle w''=6w^2+z,$
(II)$ _{\alpha}$   $\displaystyle w''=2w^3+zw+\alpha, \quad \alpha\in$$ C$

$ ('=d/dz).$ All the solutions of these equations are meromorphic in the whole complex plane $ C$ ([5], [9]). Every solution of (I) is transcendental, and equation (II)$ _{\alpha}$ admits a rational solution if and only if $ \alpha\in$$Z$ (e.g. [2], [8]); these equations define Painlevé transcendents.

The growth of a meromorphic function $ f(z)$ is measured by the characteristic function defined by

    $\displaystyle T(r,f)=m(r,f)+N(r,f)$


      $\displaystyle m(r,f)=\frac 1{2\pi}\int^{2\pi}_0\log^+ \vert f(re^{i\theta})\vert d\theta,\qquad \log^+ x=\max\{\log x,0\},$
      $\displaystyle N(r,f)=\int^r_0\bigl(n(t,f)-n(0,f)\bigr)\frac{dt}t+ n(0,f)\log r;$

here $ n(r,f)$ denotes the number of poles in $ \vert z\vert\le r,$ each counted according to its multiplicity (for the notation of value distribution theory and basic facts, see [4], [6]). Also we use the notation $ g(r)\ll h(r)$ if $ g(r)=O(h(r))$ as $ r\to\infty.$

The growth of each Painlevé transcendent is estimated as follows ([10], [11]): Theorem A. Let $ w(z)$ be an arbitrary solution of (I) $ ($resp. (II)$ _{\alpha}$$ ).$ Then, $ T(r,w)
\ll r^{5/2}$ $ ($resp. $ T(r,w)\ll r^3).$ On the other hand, Mues and Redheffer [7] have shown the following: Theorem B. For every solution $ w(z)$ of (I), we have $ \sigma(w)\ge 5/2,$ where $ \sigma(w)=\limsup
\log T(r,w)/\log r.$ By these results, the order of the first Painlevé transcendents is $ 5/2.$

In this paper we improve on the result of Theorem B, and under a certain condition, we give a lower estimate for $ \sigma(w)$ of the second Painlevé transcendents. Our results are stated as follows:

Theorem 1.1   For every solution $ w(z)$ of (I), we have

$\displaystyle r^{5/2}/\log r \ll T(r,w) \ll r^{5/2}.

An arbitrary solution of (I) is expressible in the form $ w(z)=-(u'(z)/u(z))'$, where $ u(z)$ is an entire function called a $ \tau$-function. Note that it is uniquely determined apart from the factor $ \exp(a_0z +a_1)$ $ (a_0,a_1\in$   $C$$ )$.

Theorem 1.2   For every solution of (I), its $ \tau$-function $ u(z)$ satisfies

$\displaystyle r^{5/2}/\log r \ll T(r,u) \ll r^{5/2}.

Theorem 1.3   Suppose that $ 2\alpha\in$$Z$$ .$ Then, for every transcendental solution $ w(z)$ of (II)$ _{\alpha}$, we have $ 3/2\le \sigma(w)\le 3.$

Remark 1.1   The implicit coefficients of the relation in Theorem 1.1 are estimated as follows:

    $\displaystyle \liminf_{r\to\infty} T(r,w)(r^{5/2}/\log r)^{-1} \ge 4\!\cdot\! 10^{-11}K_0^{-5}, \quad \limsup_{r\to\infty} T(r,w)r^{-5/2}\le 2K_0/5,$

where $ K_0=1+\limsup_{r\to\infty}
n(r,w)r^{-5/2}$ $ (<\infty).$

Remark 1.2   For every solution $ w(z)$ of (I), Boutroux [1] asserts the inequality $ n(r,w) \gg r^{5/2}/\log r,$ but his proof contains an incorrect part.

Remark 1.3   If $ \alpha-1/2\in$   $Z$$ ,$ then equation (II)$ _{\alpha}$ admits a one-parameter family of solutions $ \{v_c(z)\}_{c\in\text{\boldmath $C$}
\cup\{\infty\}}$ such that $ \sigma(
v_c)=3/2$ (see Section 4.3 and [2]).

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: Bibliography : Lower Estimates for the : Lower Estimates for the
Nobuki Takayama Heisei 15-9-23.