tailieunhanh - Báo cáo hóa học: "THE OPTIMIZATION FOR THE INEQUALITIES OF POWER MEANS"

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: THE OPTIMIZATION FOR THE INEQUALITIES OF POWER MEANS | THE OPTIMIZATION FOR THE INEQUALITIES OF POWER MEANS JIAJIN WEN AND WAN-LAN WANG Received 14 November 2005 Revised 5 May 2006 Accepted 14 July 2006 Let MÍí a be the tth power mean of a sequence a of positive real numbers where a a1 a2 . an n 2 and a A e Rm m 2 m 1Aj 1 min a Q max a . In this paper we will state the important background and meaning of the inequality nm 1 Mnai a j MnQ a a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained an open problem posed by Wang et al. in 2004 is solved and generalized a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization and apply techniques of mathematical analysis and permanents in algebra. Copyright 2006 J. Wen and . Wang. 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. Symbols and introduction We will use some symbols in the well-known monographs 1 5 13 An a a1 . an a a1 . an n 1 . 1 On 0 . 0 a a1 . am min a min a1 . am max a max a1 . am A A1 . Am Rn a a e R 1 i n R a a 0 1 i n R a a 0 1 i n Z a I a 0 a is a integer i 1 2 . n 0 1 n a 0 a 1 1 i n d e R Bd c a a e Rm a1 am d Bd is a finite set and it is not empty. Recall that the definitions of the tth power mean and Hardy mean of order r for a sequence a a1 . an n 2 are respectively Mnt a 1 y aA if 0 t TO n n i 1 i Mnt a J a1a2 an if t 0 Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 46782 Pages 1-25 DOI JIA 2006 46782 2 The optimization for the inequalities of power means 1 r1 Hn a r n r if r1 rn 0 Hn a r fia1a2 an if rị 0 i 1 . n where t e R r e RỊ a e R . And 1n r n c mn hn a r I Ị Ị aị a e LR r e UR is called Hardy function where i1 . in

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