218 



V— 1 ] ï — 1 



I ^a„(l— ;c-')| < — ^n I «, 

 o " 1' o 



Hence by (2) if i' > fi : 



Sa„{l-x'>)\<C~ ....... (5) 



Substitution of (1) iu Srt,,.r" gives: 



or a fortiori 



or 



S a„ X" I <' ^ — , X" , 



^ 1 ^« 



^ V V 2 



^ 2d 1— .r 



Substituting Xv^l ;- in ihe last iiietjualitj' we liave: 



I ^ «« *: i< 4 : • (6) 



From (3), (5) and (6) we deduce: 



ï— 1 « 



\ S a„ — 2 a„ X" I <^ e 



o o " 



if » >ft and .c, ^ 1 , and it follows easily that both conditions 



of our definition are satisfied. 



§ 2. 



If <,, <, . . . is an arbitrary sequence of quantities, we define the 

 so-called Holder mean-values as follows'): 



M. 0) = ^^-^ ^^ (7) 



n 



tin {t) = ... (8) 



N 



Ho\t)=H%{t) = ...... (8a) 



The following relations are easy to verify : 



//r [//"'(0]=//;f+"(o ifp>i, ?>i. .... (9) 



and 



') This definition differs slightly from the usual one, as the latter is given for 

 a series tt, + Mj + . . . and not for a sequence. 



