461 



Remark 1. 



Of course it is sufficient that the relation 1 <^ X-, < ""'"^ < k, 



is only satisfied provided ?i ^ some finite nnniber (/, since the 

 addition of a finite nnniber of terms does not influence the 

 existence or non-existence of a limit. 



Tims the function x — .»' -\- ,/•' — ^i.-' -|- x' — ;c" + ■ • • «'«es not tend 

 to a limit as x—^1 since 



. '".1+1 



^^ 3 "■•-" • - 5 



1— <^<1 — if//>5 and J — > 1 + ^ g 



1 



3 



Remark 2. 



Strictly spoken we have proved theorem 2 only if the Hoi.der- 

 mean-vaiues are limited. Now the existence of a "Holder-limit" of 

 order k implies the existence of a "Cesaró-liinit" of order k and 

 vice-versa'); hence if we prove that the Holder mean-values of 

 order ji are limited provided the Oks.akó mean-values of the same 

 order are limited, then our theorem is proved for both classes of 

 mean-values. 



Now we have (see Landau I.e.): 



Hl!"=r., T3.. TkiC;,'') (1) 



where //„ is the ?i''' Holdek meaii-valne and 6',, is the 7i''' 

 Cesakó mean-value of order k, and ; 



» — 1 .«1 4- «2 + • ■ • + 'I'll 1 

 p n p 



From (2) we deduce that x{ ^ <^c implies I l\[xï] i <^c; hence 

 it follows from (1) that | 6'* j <c implies i Hf^\ < c. 



1) Theoi'em of Knopp-Schnee. See Landau, I.e. 



30» 



