This is an announcement for the paper "Improved $L_p-$mixed volume inequality for convex bodies" by Van Hoang Nguyen. Abstract: A sharp quantitative version of the $L_p-$mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker-Csisz\'ar-Kullback inequality for the Tsallis entropy. Finally, a sharp quantitative version of the $L_p-$Brunn-Minkowski inequality is also proved as a corollary. Archive classification: math.FA math.MG Mathematics Subject Classification: 26D15, 52A20, 52A39, 52A40 Remarks: 11 pages, to appear in J. Math. Anal. Appl Submitted from: vanhoang0610@yahoo.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1506.04250 or http://arXiv.org/abs/1506.04250