432 



3^/". The coefficients a,, of the power-series for (f {t) satisfy the 

 condition 



— an for (f > 



Urn —rr-^= .... . (14) 



"''-^" 00 for (f<0 . . \ } 



11= so 



which we shall denote shortly by writing 



lïm an ^ W' ........ (14') 



>i= 00 



and by saying that the npper limit of a„ for 71 = <x> is equivalent 

 to w' . We write intentionally the sign of upper limit, becanse this 

 will do for oui' pnrpose; it is not necessary to suppose that the 

 coefficients r/„ have a "croissance régnlière". 



From the only snpposition 3a° it may already be derived that 

 the part of the earlier condition 3" corresponding to the tirst ot 

 the ineqnalities (5) is uniforinhj satisfied in the whole interval (0,1) 

 of the variable t, provided we add a factor (1— 0- ^^ prove this 

 we compare, for large values of n, the «''' derivative of (f{t) 



l\n^-2) r(n + s4-l) 



<fin){t) =,r{n\\)an^-^ -1_^ a„+i < + ...+ -^ ^- a„+,^' i .. (15) 



1 .' S; 



with the n^'^ derivative of the function 



r(A'+<f+i) 



/(0 = 



(l_i)/-l-'y-l-l 



that is 



r(A'+(f+n^l) 



r(A' + cr+w + 2) 7'(/' + rf+n + l+s) 



= l\X'^<U~n+\)+ ^ \, ^ +-+ , ^^ - + ...(16) 



1 / s! 



Here d is a certain positive number. However small we may 

 choose this, there is always, on account of our condition 3(^ï" corre- 

 sponding to any arbitrarily small number f an integer N such that 

 for all values of 5 > 



a„_|_, < f — — -— 



r(n + «-fl) 



Thus we have uniformly in the interval < ^ <^ 1 



^ — ^-^<?, for nyN . . . (17) 



r(yl' + (f-fw-|-l) ^ ^ ^ ^ 



here q){t) means the natural majorant of (p [t) determined by 

 (f{t) =^ [aj + |aj « 4- . . . + \an\ «" + ... 

 The same inequality a fortiori holds for (f{t) itself, and since <S 

 can be taken arbili-arily small, it follows that the integral (11), 



