We first discuss some relations between positivity, geometric PDEs and Hodge-index type theorems. It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials inspires a generalized form of Hodge index inequality. Inspired by this result, we obtain a mixed Hodge-Riemann bilinear relation. The new feature is that this Hodge-Riemann bilinear relation holds with respect to classes in which some satisfy particular positivity condition, but could be degenerate along some directions. We also discuss its some connections with the Hodge theory of algebraic maps.