Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : On Minkowski’s inequality and its application | Zhao and Cheung Journal of Inequalities and Applications 2011 2011 71 http content 2011 1 71 3 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access On Minkowski s inequality and its application Chang-Jian Zhao1 and Wing-Sum Cheung2 Correspondence chjzhao@163. com department of Mathematics China Jiliang University Hangzhou 310018 China Full list of author information is available at the end of the article Abstract In the paper we first give an improvement of Minkowski integral inequality. As an application we get new Brunn-Minkowski-type inequalities for dual mixed volumes. 2000 Mathematics Subject Classification 52A30 52A40 26D15 Keywords Minkowski s inequality H o lder s inequality Brunn-Minkowski inequality dual mixed volume 1 Improvement of Minkowski s inequality The well-known inequality due to Minkowski can be stated as follows 1 pp. 19-20 2 p. 31 Theorem Letf x g x 0 and p 1 then with equality if and only if f and g are proportional and if p 1 p 0 then with equality if and only if f and g are proportional. For p 0 we assume that f x g x 0. An almost improvement of Minkowski s inequality for p e R 0 is obtained in the following Theorem Theorem Letf x g x 0 and p 0 or f x g x 0 and p 0. Let s t e R 0 and s t. Then i Let p s t e R be different such that s t 1 and s - t p - t 1. Then X f x g x pdx 1 s s p-t s t 1 tlt s-p s-t Springer with equality if and only if fix and g x are constant or 1 p 1 s 1 t 2 andf x and g x are proportional. ii Let p s t e R be different such that s t 1 and s t 0 and s - t p - t 1. Then 2011 Zhao and Cheung licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http licenses by which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Zhao and Cheung Journal of Inequalities and .
