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# Introduction

Consider the first and the second Painlevé equations

 (I) (II)

All the solutions of these equations are meromorphic in the whole complex plane ([5], [9]). Every solution of (I) is transcendental, and equation (II) admits a rational solution if and only if $Z$ (e.g. [2], [8]); these equations define Painlevé transcendents.

The growth of a meromorphic function is measured by the characteristic function defined by

with

here denotes the number of poles in each counted according to its multiplicity (for the notation of value distribution theory and basic facts, see [4], [6]). Also we use the notation if as

The growth of each Painlevé transcendent is estimated as follows ([10], [11]): Theorem A. Let be an arbitrary solution of (I) resp. (II) Then, resp. On the other hand, Mues and Redheffer [7] have shown the following: Theorem B. For every solution of (I), we have where By these results, the order of the first Painlevé transcendents is

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

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

An arbitrary solution of (I) is expressible in the form , where is an entire function called a -function. Note that it is uniquely determined apart from the factor    $C$.

Theorem 1.2   For every solution of (I), its -function satisfies

Theorem 1.3   Suppose that $Z$ Then, for every transcendental solution of (II), we have

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

where

Remark 1.2   For every solution of (I), Boutroux [1] asserts the inequality but his proof contains an incorrect part.

Remark 1.3   If    $Z$ then equation (II) admits a one-parameter family of solutions such that (see Section 4.3 and [2]).

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