We prove that a function $f$, analytic in the unit disc, belongs to the Hardy space $H^1$ if and only if $$ \sum^n_{j=0} \frac1{+1} \|s_j f\| = O (łog n) \quad (n\to\infty), $$ where $s_jf$ are the partial sums of the Taylor series of $f$. As a corollary we have that, for $f\in H^1$, $$ \sum^n_{j=0} \frac1{j+1} \|f-s_jf\| = o(łog n), $$ The analogous facts for $L^1$ do not hold.