Exact estimates for moments of random bilinear forms

The present paper concentrates on the analogues of Rosenthal's inequalities for ordinary and decoupled bilinear forms in symmetric random variables. More specifically, we prove the exact moment inequalities for these objects in terms of moments of their individual components. As a corollary of these results we obtain the explicit expressions for the best constant in the analogues of Rosenthal's inequality of ordinary and decoupled bilinear forms in identically distributed symmetric random variables in the case of the fixed number of random variables.