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

In the paper, the concept of Lp-dual three-mixed quermassintegrals is introduced. The formula for the Lp-dual three-mixed quermassintegrals with respect to the p-radial addition is proved. Inequalities of Lp-Minkowski, and Brunn-Minkowski type for the Lp-dual three-mixed quermassintegrals are established. The new Lp-Minkowski inequality is obtained that generalize a family of Minkowski type inequalities. The Lp-Brunn-Minkowski inequality is used to obtain a series of Brunn-Minkowski type inequalities.

In this paper we present some results about k-perfect numbers, and generalize two inequalities due to M. Perisastri (see [6]).

In the paper, our main aim is to generalize the qth dual volume to Lp space, and introduce pqth -dual mixed volume by calculating the first order variation of qth dual volumes. We establish the Lpq-Minkowski inequality for pqth-dual mixed volumes and Lpq-Brunn-Minkowski inequality for the qth-dual volumes, respectively. The new inequalities in special case yield some new dual inequalities for the qth-dual volumes. [ABSTRACT FROM AUTHOR]

In the paper, the concept of Lp-dual three-mixed quermassintegrals is introduced. The formula for the Lp-dual three-mixed quermassin-tegrals with respect to the p-radial addition is proved. Inequalities of Lp-Minkowski, and Brunn-Minkowski type for the Lp-dual three-mixed quermassintegrals are established. The new Lp-Minkowski inequality is obtained that generalize a family of Minkowski type inequalities. The Lp-Brunn-Minkowski inequality is used to obtain a series of Brunn-Minkowski type inequalities. [ABSTRACT FROM AUTHOR]

In this short note, we give a generalization of Berker's inequality with one parameter in En by using the theory of Schur-concave function. [ABSTRACT FROM AUTHOR]

In this short note, we give a generalization of Berker's inequality with one parameter in En by using the theory of Schur-concave function. [ABSTRACT FROM AUTHOR]