Inequalities for pq^th-dual mixed volumes

In the paper, our main aim is to generalize the q^th dual volume to L_p space, and introduce pq^th-dual mixed volume by calculating the first order variation of q^th dual volumes. We establish the L_pq-Minkowski inequality for pq^th-dual mixed volumes and L_pq-Brunn-Minkowski inequality for the q^th-dual volumes, respectively. The new inequalities in special case yield some new dual inequalities for the q^th-dual volumes.