ABSTRACT
The first conclusion to be drawn from this inequality is that (4.04) holds in (0, X,] . A second conclusion is that X,+, > X,; compare (1.16).t Thus 2, = X,. And i f x > X,, then E,(x) = l and Ev+ ,(x) 2 1. Accordingly, (4.04) also holds in (X,, co) and the Iemma is proved. 4.2 Theorem 4.1 With the conditions of $2.1, equation (2.05) has, for each value of u and each nonnegative integer n, solutions Wzn +,, (u, C ) and W2, + , , (u, C ) which are repeatedly differentiable in the C interval (a, O), and are given by
ICI BJC) - - J ~ + I I I ~ ~ ~ + Z ~ + , , , u (4.05)
where
The proof is similar to that of Theorem 3.1. Writing
C now being positive, and
Using (1.17) and the fact that E , ( u u ' ~ ~ ) is a nonincreasing function of u, we derive IK(C,u)l < nC-v12v(v+1)12E-1( , ur1l2) E,(uv'I~) M,(uc'") M ~ ( u v ~ ~ ~ ) (0 < v < 0;
(4.1 2) compare Chapter 1 1 , (3.14). The derivative of the kernel is given by
= +au(v/~)(v+1)12Ev+1(~C112)E~2(~~112) Mv+1(~C112) x E , ( u v ~ ~ ~ ) M , ( u v ~ ~ ~ ) Tv(C,v), (4.13)
where
Use of Lemma 4.1 and the fact that E , ? ( u [ ~ / ~ ) E ; ~ ( u ~ ) ' / ~ ) d 1 shows that
