Let f be a conditionally symmetric martingale and let S( f) denote its square function. (i) For p, q > 0, we determine the best constants C_(p,q) such that sup E(|f_n|~p)/((1+S_n~2(f))~q)≤C_(p,q). Furthermore, the inequality extends to the case of Hilbert space valued f. (ii) For N = 1,2, ... and q > 0, we determine the best constants C_(N,q)~' such that sup E(f_n~(2N-1))/((1+S_n~2(f))~q)≤C_(N,q)~' These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if the conditional symmetry is not assumed.
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