(eg) 
Oo 
oP 
The conditions 
F (u) = Oru"), and f(v) = Ov”), 
mentioned in § 1 and at the beginning of § 2 give the relations 
F (u) = O (u?) = O (ui) 
Ke PV E u \ 
pl ee p / p” 
and 
/u 
= Out) 1 
Pe 
= O (u?). O (log u) 
= 0 (u) ; 
hence it follows that the function considered is in this case equal to 
O (uw) — Ou”) = O(w'n). 
According to the first lemma we. have 
1 
: ‚m 
z [roo Le (5) =0 en, 
u=2 1 m P ( og &) 
u 
1 
TEK en if U am 
Fw EE fF {| |) Ol . 
u=2 u=2 m/ p (log #) 
u—l u—l P uM 
f 1 
; : am 
Se gd Clty a 
pr vou (log x)? 
(Le vl 
and according to the last lemma this may be modified to the 
formula sought 
1 1 
x É AX in Ek av mt ; 
Se) = 0 se a TOT =de 
5 2 
u—? log U (log 2) ’ 
§ 3. By starting from formula (1) by which the mean value of 
the function /’(w) has been given in the interval from 1 to « (the 
limits included), it is possible, as is known, to determine in an 
elementary way the mean value in the same interval of a number 
of other functions, connected with the function /’; this we shall 
however only elaborate for some cases. 
Lemma. From (1) ensues 
