$q$-Quermassintegral forms of the $L_p$-Busemann–Petty centroid inequality

Lutwak, Yang and Zhang established the $L_p$-Busemann–Petty centroid inequality. We establish the $q$-quermassintegral forms of the $L_p$-Busemann–Petty centroid inequality for convex bodies. As special cases, we obtain the quermassintegral and harmonic quermassintegral versions of this inequality for convex bodies.