DUNKEL. LINEAR DIFFERENTIAL EQUATIONS. 361 



t=l—L 



(-) p^p "| / s i * i • • ■/ i i & \J B M " i ■<* * r* i d * 



t=l—L 

 < 



t integrations 

 C 



V I log x K- ] r , ifg '_ / - fv? | log a; j e «~ A dx (Lemma VII.) 



JS log X \ L .) J 



t-l-L p 



< J/ 91 2 I lo S x l'- 1 -''- 7 -' / B I log x | e < _A rfa 



c 



< M* #- L } fB I log x l^- 1 dx < M q ^' V-^ ■ 

 — lo<£ x \ J L- 



o 

 Therefore : 



Vfc, 7, </!+! I — \C 



-L 



~C~ 





The limit of the left side of (54) for x = is zero by Lemma VI., 



■ ■ 



while the same thing is true of the left side of (55) from the inequali- 

 ties. Therefore (43) is true in this case for q — q l + 1. 

 From (33) we have for case (e) : 



(56) 



ia, +1 |<_L_I l'-\dx\... f-\dx\ f l -M q ^M(x)\dx\ 

 1 v ' ,qi+ ' - | loga;|'- A [_.) x l J ar 1 'J x C v ' ' 



C ' c 



(£ — Z,) integrations 



X XX 



+ 2 f-|<&|... f-\dx\ fBM Ql \\ogx\ e *-*\dx\ 1. 



c c c 



< integrations 



In the first part of the bracket we have used the inequality obtained 

 from (52) : 



Now (he only difference between the inequalities (53) and (50) is in 

 the power of | log x | outside the brackets ; and, since | log x \ 

 | log x V~ L , all the results that we have- obtained from (53) will follow 

 also from (56). Then for all the sub-cases of II., (43) and (48) are 

 true for q = q x -f 1. 



