354 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Therefore, when e approaches zero, the integral on the left side of all 

 these inequalities converges, and we have : 



X X 



1.4+1 w 1 = IJ'Ia w dx I =/' b 1 1 lo s x \ k *"• 



6 



Thus the two lemmas are proved when x < 1 ; and it is easily seen how 

 to conclude the proof of I. in case c > 1. 



Lemma III. If b is a continuous function of x in the interval 

 < x ^ c, and its absolute value is integrable up to x = 0, ane? i/\* 



X XX 



F t (x) = I -dx . . . I -dx I x r bdx 



00 



t integrations 



where r is real and greater than zero, then : 



x 

 (37) \x-rF,(x)\^-Lj'\b\dx. 





 When t = 1 we have : 



x xx 



I af-- F x (x) I = x~ r \ I x r b dx \< x~ r I x r \b\dx < I \b\ dx, 

 60 



and so in this case III. is true. Assume that it is true for t = t x ; then 

 it is also true for t = t 1 + 1. For: 



X X 



__„ , ., _i n „ ... , 



X 



F^ (X) I = X- \f l - F^ (X) dx I < X-'f 1 - \ F^ (x) | dx 



6 



x r r x -1 



00 J 



Therefore III. is true for all values of £. 



