590 



j ^{ny '{t— H)~'-'dn < g j m«-1(« - ?<)- 



du 



i ^ " J 



o o 



01', substituting Tl = tv 



1 1 



o ' " 



so that tlie integrul in llie left-liaiid member of this inequality 

 remains finite for all values of / in the closed interval (0,1). F'urther 

 vv^e divide the second integral on the right-hand side of (28) as 

 follows, su|)posing t'^h, 



j i[»'(u)u''(t— ft) ^^ du = j f I 



(I <— (1 -t) 



To the tirst of these two integrals we apply the second mean 

 value theorem, which is allowed, because the expression n'-^ {t — ?t)~=' 

 increases monotonously in the interval in question. We obtain 



ip'(M)u«(<— u) ■^- da = {2t^~lY{]—ty-< [»|>(2<- 1 )— iKf?)] 







where {i is a number in the inlei-val (0, 2t — 1). This part of the 

 integral, as (/ it) remains within finite limits, is therefore for t = 1 

 at most ecjuivalent to (1 — 1)'~'-'-. In order to infer the same thing 

 with regard to the second integral, we make use of the fact that 



lifn{\—t)"xpi")(t) — 0, (/J ^1,2,...) . . . . (29) 



t=i 



We shall prove this at once; it should not be thought that it is 

 a consequence of the proposition mentioned in a footnote of ^ 1 : 

 it follows solely from the convergence of the series (27) for /= 1. 



If we assume, for a moment, the formula to be true, we have 

 for the whole interval 0<«<1, if K is a certain positive number, 

 not depending on u, 



K 



1 — M 



and so, in the interval of integration '2t — 1 < n < t 



K 



ƒ 



"•;<"' <r3r 



from which it follows 



