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Inequalities of the form $$\sum\nolimits_{k = 0}^\infty {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $$ for all$f$∈$H$^{1}, where {$m$_{$k$}} are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterize$B$-convexity. It is also shown that for 1<$q$<∞ and 0<α<∞ the space$X=H$(1,$q$,γa) consisting of analytic functions on the unit disc such that $$\int_0^1 {(1 - r)^{q\alpha - 1} M_1^q (f,r) dr< \infty } $$ satisfies the previous inequality for vector valued functions in$H$^{1}($X$), defined as the space of$X$-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {$m$_{$k$}}={2^{k}} but not for {$m$_{$k$}}={$k$^{$a$}} for any α ∈ N.

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