32 -^rt- 4. — M. Euniyecla 



(7" + n' 



(32) / = - 



/ 



From the relation a'y<,\, we obtain 



Hence --j—^ "^ r > '^' 



since a" > and o > 0. Tlierefore the right-hand side of (32) is 

 ultimately negative and hence there is no stationary value of ^. 



A 

 If /o = 0, then f = yip^' 



and evidently there is no stationary value of <p. 



We easily see that, as x-^0, (p tends to zero by negative 

 values, and that, as x^d from below, ^^ — co and, as x->d from 

 above, (p^ + ^ . Thus in the case (i), the function ^ is a steadily 

 decreasing function of x in the interval 0<ic<«, except for the 

 value x = d, and it is a steadily increasing function of x in the 

 interval « < a- < 1^, the stationary value for x = a being plainly a 

 minimum. In the case (ii), <p is a steadily decreasing function of 

 X throughout the interval < a; < ç, except for the value x = d. 



Evmeuiiy s^ is continuous throughout the interval 0<:x<$, 

 except for x = 0. 



The proof of the lemma is thus completed. 



18- I will give other lemmas of a different type. 



Lenima 4. Lci/(y) and }\{y) be L-functions snch that 



y* >f(y) > (Vyy> y >Uy) > (^lyY> 

 My)-^f(y)>^^ 



as î/->cc . Jf y=6 and y='di are respectively the roots of the equations 



yfiv) = ^> yfiiy) = c^, 



for large indues of /, c being a, positive constaid independent of ^, then 



