A multilinear form inequality

An inequality of Interpolation type for Multilinear Forms with a two-part dependence condition is proved. It generalizes the work of Bradley and Bryc [Theorem 3.6, Multilinear forms and measures of dependence between random variables, J. Multivariate Anal. 16 (1985) 335–367] and Prakasa Rao [Bounds for rth order joint cumulant under rth order strong mixing, Statist. Probab. Lett. 43 (1999) 427–431].