360 PROCEEDINGS OF THE AMERICAN ACADEMY. 



since the greatest value of e K — A + t — 1 is 



e K — X + L — 1 = e K - X + {e K — e K + X) - 1 = e K — 1. 



X 



(51) | ^ z> fc+1 1 < M q 'LjB J log z l^- 1 <fe = Jf* ^ JIf (*) 



o 



< M q 'M(x) < M qi+1 . 



It is obvious that the above result holds also for II. (c) (31). From 

 (32) we shall obtain in the same way for II. (d) : 



X 



(52) | log x \ l ~" | <f> KMi+1 1 < M* LJB I log x \ e ^ l dx = M* ^ M(x) 



o 



U , , I < , x - Mix) < M qi Mix) < M gi+ \ 



In all three sub-cases (or), (c), and (d), it is easily seen that (43) is 

 true for q — q± + 1. We have now left of case II. the sub-cases (b) 

 and (e). 



From (30) we have for case (b) : 



I**, 4_il^ n 1 uA f-\dx\... f^\dx\f-M q ^~M(x)\dx\ 



c c c 



(< — L) integrations 



t=l-L x * x -, 



+ 2 f-\'<k\...f-\dx\lBM qi \logx\ e *- x \dx\\. 



C C 



t integrations 



In the first part of the bracket we have replaced | <f> k ,L, qi +l I by 

 M Ql -^1 M (x), using inequality (51). 



We will consider the two parts of (53) separately. 



(54) xxx 



l — T f-\dx\... f-\dx\ f-M q ^M(x)\dx\<M q ^M(c) 

 x\'- L J x 1 J x J x U O 



log 



c c c 



{I — L) integrations 



_ M qi+1 - (Lemma VII.) 



G 



