PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 58(72), pp. 149152 (1995) 

A remark on the partial sums in Hardy spacesM. Pavlovi\'cMatematicki fakultet, Beograd, YugoslaviaAbstract: We prove that a function $f$, analytic in the unit disc, belongs to the Hardy space $H^1$ if and only if $$ \sum^n_{j=0} \frac1{+1} \s_j f\ = O (\log n) \quad (n\to\infty), $$ where $s_jf$ are the partial sums of the Taylor series of $f$. As a corollary we have that, for $f\in H^1$, $$ \sum^n_{j=0} \frac1{j+1} \fs_jf\ = o(\log n), $$ The analogous facts for $L^1$ do not hold. Classification (MSC2000): 30D55 Full text of the article:
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