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 Schur-Convexity of Two Types of One-Parameter Mean Values in n Variables | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 78175 10 pages doi 2007 78175 Research Article Schur-Convexity of Two Types of One-Parameter Mean Values in n Variables Ning-Guo Zheng Zhi-Hua Zhang and Xiao-Ming Zhang Received 10 July 2007 Revised 9 October 2007 Accepted 9 November 2007 Recommended by Simeon Reich We establish Schur-convexities of two types of one-parameter mean values in n variables. As applications Schur-convexities of some well-known functions involving the complete elementary symmetric functions are obtained. Copyright 2007 Ning-Guo Zheng et al. 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 Throughout the paper R denotes the set of real numbers and R denotes the set of strictly positive real numbers. Let n 2 n G N x x1 x2 . xn G R and x1 r x r x2 r . x1 r where r G R r 0 let En-1 c Rn-1 be the simplex E. n-1 Ị 1 Un-1 Ui 0 1 i n - 1 n-1 Ui 1 i 1 and let dp du1 . dun-1 be the differential of the volume in En-1. The weighted arithmetic mean A x u and the power mean Mr x u of order r with respect to the numbers x1 x2 . xn and the positive weights u1 u2 . un with y n 1ui 1 are defined respectively as A x u y n 1uixi Mr x u y n 1uixrị for r 0 and M0 x u n n 1xUi. For u 1 n 1 n . 1 n we denote A x u A x Mr x u Mr x . The well-known logarithmic mean L x1 x2 of two positive numbers x1 and x2 is x1 x2 L x1 x2 1 lnx1 - lnx2 x1 x2 x1 x1 x2. 2 Journal of Inequalities and Applications As further generalization of L x1 x2 Stolarsky 1 studied the one-parameter mean that is 1 - x2 1 Y r 1 X1 -X2 r - 1 0 X1 X2 X1 - X2 Lr x1 X2 ln x1 - ln x2 ự xx 1 X1-X2 Ạxỹ X1 r -1 X1 X2 r 0 X1 X2 X1 X2. Alzer 2 3 obtained another form of one-parameter mean that is Fr x1 X2 r XĨ 1 - x2 1 ----- - . r 1-x - x2 r - 1 0 X1 X2 ln X1 - ln X2 .
