282 



therefore the same for all functions satisfying the assumptions of 

 the theorem). 



Hence, if Mn{Q) is the maximum value of |Q„(.r„)| on the circle 



I ;r„ \ = o (?i =; 1, 2, . . . ), we have r„ ^ k M,, ((/). Since we have 



00 00 



proved that ^ r,, is convergent, it now follows that the series ^ Af„((>) 



11=1 "=i 



converges also (for p<^l). From this fact llie theorem of ^1 can 

 be easily deduced. 



For let ft, {x„) = 1 ötp' J!, {n = 1,2, . . .). Then 



''^^/« = 1,2...A 



If z= 6 G (where 6 is the constant of ^ 1), then it follows 

 that, if <[()<[ 1, ( we take for example (> :^ — - — j, 



2 M,{q) 



2 |a(n)| @P< 

 ,,=i' /- ' - 1-0 



Hence the series 



2 :S I at") I 0'' , 

 „=i,,=i P ' 



is also convergent. This proves a fortiori the convergence of the 



given |)Ower-series in an i. n. of v. for j,i,'„[ ^ = (9 Cr(n = J , 2 .. .). 



It cannot be denied that the assumption, that <i is an integral function, 



is somewhat unaesthetic. However, the author has nol succeeded 



in dealing with the more general problem, where (p is an arbilrarj" 



(purely formal) power-series. In any case the method described does 



not give the required result in the more general case. 



Copenhagen, November J 922. 



