Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On an Extension of Shapiro’s Cyclic Inequality | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 491576 5 pages doi 2009 491576 Research Article On an Extension of Shapiro s Cyclic Inequality Nguyen Minh Tuan1 and Le Quy Thuong2 1 Department of Mathematical Analysis University of Hanoi 334 Nguyen Trai Street Hanoi Vietnam 2 Department of Mathematics University of Hanoi 334 Nguyen Trai Street Hanoi Vietnam Correspondence should be addressed to Nguyen Minh Tuan tuannm@ Received 21 August 2009 Accepted 13 October 2009 Recommended by Kunquan Lan We prove an interesting extension of the Shapiro s cyclic inequality for four and five variables and formulate a generalization of the well-known Shapiro s cyclic inequality. The method used in the proofs of the theorems in the paper concerns the positive quadratic forms. Copyright 2009 N. M. Tuan and L. Q. Thuong. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In 1954 Harold Seymour Shapiro proposed the inequality for a cyclic sum in n variables as follows X1 I X2 I I xn-1 I Xn n X2 X3 X3 X4 Xn X1 X1 X2 2 where Xi 0 Xi Xi 1 0 and Xi n Xi for i e N. Although was settled in 1989 by Troesch 1 the history of long year proofs of this inequality was interesting and the certain problems remain see 1-8 . Motivated by the directions of generalizations and proofs of we consider the following inequality X1 X2 Xn-1 Xn P n p q ---- ----- -------- ----- ----- ---- pX2 qX3 pX3 qX4 pXn qX1 pX1 qX2 n p q 2 Journal of Inequalities and Applications where p q 0 and p q 0. It is clear that is true for n 3. Indeed by the Cauchy inequality we have x1 x2 x3 2 4 -x1----A x1 px2 qx3 4 --x2---A x2 px3 px2 qx3 px3 qx1 I------- --------------K 2 1 lx3 px1 qx2 px1 qx2 P 3 p q p q x1x2 x2x3 x3x1 . It follows that P 3v a x x2 x3 2 3 p q x1x2 x2x3
