Mathematics. — "On n Genei'alisdt'um of Taubek's Theorem con- 

 cerning Power Series". By M. J. Belinfante. (Coinmunicated 

 by Prof. L. E. J. Brouwer). 



(Communicaled at the meeting of March 24, 1923). 



Introduction. 



In iliis paper we conside,r power series with complex coefficients, 

 bnt for real values of the variable. We suppose them to converge 

 if \x\ <^1, and we denote by ,r — > 1 that x approaches 1 by reaJ 

 values from below. 



Tauber has proved the following theorem'): 



If lim.na„^:0 and /im. l£a„x" ^ .?, then 2 a,, converges to .i. 



n=oo X — ^1 



LiTTLEWOOD ') has shown that the usual proof of this Iheorem 

 proves more than is actually stated, and that the same proof applies 

 to the theorem ; 



00 



If 2 anx" oscillates finitely as x—^1, then the limits of oscillation 







» 00 



as 71 —*■ 00 of 2 ai are the same as the limits of oscillation of ^ a„a;". 







In the present paper we give extensions of both theorems to the 

 so-called mean-values of Holder. 



§ 1 contains the proof mentioned above and a definition of the 

 expression "oscillate about the same region"; in ^ 2 the definition 

 of Holder's mean-values and some necessary formula's will be treated, 

 while § 3 contains the generalizations of Taüber's theorem. 



§ t- 



Definition^). We say that /(x) oscillates for x—^.t^ about the same 

 region as g{y) for y—*y„ when the following conditions are satisfied : 



>) Monatshefte fur Matli. u. Phys., 1897 Bd. 8, p. 273. 

 ') Proc. of the Lend. Math. Soc. 1911 Vol. 9, p. 436. 



') We always suppose that ,«; resp. y approaches .Cq resp. tj^ by real values 

 from below. 



