Introduction

Consider the first and the second Painlevé equations

$$w''=6w^2+z,$$ (I)

$$_{\alpha} w''=2w^3+zw+\alpha, \quad \alpha\in C$$

$('=d/dz).$ 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

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

with

$$m(r,f)=\frac 1{2\pi}\int^{2\pi}_0\log^+ \vert f(re^{i\theta})\vert d\theta,\qquad \log^+ x=\max\{\log x,0\},$$

$$N(r,f)=\int^r_0\bigl(n(t,f)-n(0,f)\bigr)\frac{dt}t+ n(0,f)\log r;$$

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 $\sigma(w)=\limsup _{r\to\infty} \log T(r,w)/\log r.$ 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

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

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

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

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:

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

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]).