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F'or shortness' sake we confine ourselves lo the fust case. (The 

 proof in the second case is not essentiall}' ditferenl, though in details 

 more intricate). Tlien the functions Q„{x„) are, because G ^ 1, ail 

 regular in their rosp. circles |,c„|^l. 



For any function ƒ (s), regular for [^|<^ 1, and for which /(O) =:0, 

 we now define a number r as follows: ?• is the radius of the largest 

 circle, of which all points represent numbers assumed by f (z) in 

 the circle J2|^ 1. Let ?•„ (7i = J , 2, . . .) be the corresponding quantity 



for Q,t {Xu). Then we first prove, that the series 2 r„ converges. 



For this purpose we consider (4), valid for all sets of values of 



,1',, .1', x,n, satisfying |.i'„[ ^ 6r^ (/t = i, 2, . . . . m), und, a fort ioi-i, for 



all satisfying j.r„[<^l. Because fp U/) is an integral function, it is 

 possible to choose a number L so large, that the maximum value of 

 \if'{y)^, on the circle \y\^ L, is ^ K. Now suppose that, for some 

 value of m, r, -|- >■,-)-.... -}- 7',„ ^ L. Then the maximum value of 

 \if'{y)^ on the circle |iy;^ 'i + '', + •••■+ ''»' would be ]> K. 

 Now if we let the variables x„ (?; = d , 2, . . . . m) describe their resp. 

 circles ].ï„j^ 1, then Q,, (.(■„) assumes all values satisfying | Q« fei)! ^ '""• 

 Hence y =z Q^ (x) -)- Q, (x^) -\- . . . . -\- (2,n {.v,„) assumes all values 

 satisfying \y\ ^ i\ -\- r, -\- . . . . -\- r,„. Therefore it would be possible 

 to find a set of values x\,x\, (.',„ such that 



.'/ ^ Q> (*',) + Q. i^',) + • • • + Q.« (*•'.«) -- ('•, +'•, + • • + r„,)e'^, 

 where (r, -\- 7\ -\- . . . . -\- ?•„,) «;'^ represents that point of the circle 

 j(/|^= ^'i + '', "t- • ■ ■ + >'m where | (p (y) \ assumes its maximum value. 

 Therefore we should have 



I (f (Q, (^\) + Q. K) + • • . ^ Q,n(-^',n)) 1 > K, 



contradictory to (4). Therefore the supposition ?',-f-'',+ • • • + '"m ]>-^ 

 can not be true. Since L is independent of m, this proves the 



convergence of S ?•„. 



We now apply the following theorem of Bohr'): 



CO 



Let the function /{z) = ^ a„ c" {ƒ (0) = 0) be regular for | 2 | ^ 1. 



Let M{q) be the maximum value of 'f[z)i on the circle \zI=q 

 (0<^p<[l). Then, if r is the quantity defined above, we have 

 r^kM{Q), where k is a number which depends on p only [k is 



') Not yet published. 



