(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

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:

where