Abstract. Our main result is a generalization of a Paley type inequality. Namely, the estimation $$\big(\sum_{n=0}^\infty m_n^{1-2/p}M_n^{2-2/p} \sum_{j=1}^{m_n-1}|{\hat f}(jM_n)|^2\big )^{1/2} \leq C_p\|f\|_{H^p_{**}} (*)$$ is proved for martingales $f\in H^p_{**}(G_m)$ ($0< p\leq1$), where $\hat f(.)$ denote the Vilenkin--Fourier coefficients of $f$ and the Hardy space $H^p_{**}(G_m)$ is defined by means of a maximal function. <br /> We formulate the dual inequalities of $(*)$ and its variant for other Hardy- and ${\cal BMO}$-spaces, too. The so called ``bounded case" is also investigated, specially as a corollary we get the known generalization of Khinchin's inequality in this case.

AMS Subject Classification (1991): 42C10, 60G42

Received March 8, 1996 and in revised form November 11, 1996. (Registered under 6106/2009.)