DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 353 



tegrable up to x = 0, then b can be integrated t times from x = as 

 follows : 



X XX 



(35) /,(*)= C-dx . . . / -dx I bdx. 



o oo 



t integrations 



Lemma II. If the conditions of Lemma I. hold, then : 



(36) \f(x)\ = I C-dx. . . C-dx Cbdx\< I \b\ | log x |'- dx. 



o o o o 



t integrations 



(0 <x < 1). 



We will prove these lemmas by mathematical induction. They are 

 true when t = 1. Let us assume that they are true for a particular 

 value of t, say t = t x . 



Let X be any particular value of x in the interval < x < 1, and 

 choose e at pleasure such that < e < X. Then we have : 



X X X x 



I j; A (*)<** I = Jl'A < x ) I *** =j\ dx J\ bl |log *f 1-lrfx 



■ € € 



= log X C\b\\ log a: f 1-1 rfx - log e f| 6 | | log x |' 1_l dx 

 o o 



— / log x I & I I log x |' 1_1 dx 

 = I tog e I f I b I I lo g * I' 1 " 1 dx+ C\b\\logx I* 1 rfx 



€ 



, -llogXI^I^IIlogxl' 1 - 1 ^ 



6 

 * x 



< | log c | A b 1 1 log x \ h ~ l dx +f\b\\ log x |* rfx 



< 



€ X X 



< C\ b I I log x f l dx + C\ b | | log x |* <fe = f | b | | log x | fl rfx. 







vol. xxxviii. — 23 



