In this paper, applying for the Minkowski's and Hölder's integral inequalities, we obtain four theorems about the (p, q)-mixed volume involving the L p centroid bodies and the L p intersection bodies, respectively. The former two theorems reveal the convexity of the functionals related to the (p, q)-mixed volume, in terms of the dual Blaschke addition introduced in [Journal of Geometric Analysis, 30 (2020) 3026-3034], and the latter two theorems expose the monotonicity of the other functionals related to the (p, q)-mixed volume.