74 Mr Kienast, Proof of the equivalence of 



Proof of the equivalence of different mean values. By Alfred 

 Kienast. (Communicated by Professor G. H. Hardy.) 



{Received 12 April: read 8 May 1920.] 



If Oi, ao, ... an, ••• denote the terms of a sequence of complex 

 numbers, and 



8^^ = a, + ... + an, 



then lim 8^"^ i [ ) is called Cesaro's /cth mean* of the se- 



:.(0) 



quence 8\ 



Putting h^^^= ai + . . . + a^, 



A<:'=ii/,«;'+...+An, 



then lim A^^' is called Holder's /cth meanf of the sequence hj. 



In a paper "Extensions of Abel's Theorem and its converses |" 

 I found it convenient to introduce the expressions 



5f=ai-F...+a,^ (?i=l, 2, ...), 



^'^-l[sT^-+sf_,] (.. = 2,3,...), 



and proved various theorems concerning the limits lim s^ . 



Several writers have proved 



Theorem 1. Whenever Cesaro's (Holders) Kth mean exists 

 and is finite, then Holders (Cesaro's) /cth mean eooists too, and both 

 have the same value. 



* Bromwich, Infinite series, p. 310. t Ibid., p. 313. 



t Proc. Cambridge Phil. Soc. , vol. xix, 1918, pp. 129-147. 



