Some subsequent papers in much the same vein are due to G. Halasz [3] and to J. M. Anderson and K. G. Binmore [1]. Halasz has obtained estimates for ani 1 an, and Z1 IanI when An + l/Anq> 1 and f satisfies appropriate conditions. The paper of Anderson and Binmore is concerned with the application of coefficient estimates to the study of entire functions. Their work includes the estimate lanI _ (2A,n)112pn Ifl 2 where Pn is as above, 2 A1 < oo, and fe L2(0, so). L. Schwartz [8] had earlier obtained results which essentially include both the estimates of Gaier and of Anderson and Binmore. Indeed, as the referee has pointed out, S. Mandelbrojt [5] had also established inequalities of the same kind and these appear in larger form in [6]. Later, Mandelbrojt's results were extended by F. Sunyer Belaguer [9], [10]. There is some question left to the sharpness of Schwartz's estimates and in ?2 it is proved that if 1 ?p ?2 and fe LP(O, oo) then