326 UNIVERSITY OF VIRGINIA PUBLICATIONS 



The quotients on the left are all positive, the least and the greatest are 

 respectively the first and the last, and are 



M = - ~ 



M 



h 



^ (xi) - xp jxo) _ ■•p jh) 

 h h 



m< 2V2- <nM, 



,<?^<M. 



Also 



Whence for all small values of x = ^h, 



271^^'"-^)<7T^^272^(^^- 

 Since ^(jtt — x)> <p(|7r) and v'(.^)>0, there results 

 2V2 ^^ xkix) ^4: 



IT XT 



and consequently ^{x)/x remains finite and positive as x converges to zero. 

 Moreover 



4'i.x) fi^^'P^i^^ 

 = <P Kix) -J- — 



Since 0<.p(|a;)<l, 



^X X ' 



and the sequence of increasing numbers 



^ fa/2) i(x/2^) 

 x/2 ' x/2^- ' 



