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: A RECONSIDERATION OF HUA’S INEQUALITY. PART II HIROYUKI TAKAGI, TAKESHI MIURA, TAKAHIRO | A RECONSIDERATION OF HUA S INEQUALITY. PART II HIROYUKI TAKAGI TAKESHI MIURA TAKAHIRO HAYATA AND SIN-EI TAKAHASI Received 20 April 2006 Accepted 16 May 2006 We give a new interpretation of Hua s inequality and its generalization. From this interpretation we know the best possibility of those inequalities. Copyright 2006 Hiroyuki Takagi 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 In 1965 L. Keng Hua discovered the following inequality. Theorem 2 . If 8 A 0 and x1 . xn e R then n 2 n 2 8 - ị xj A ị xi A -. i 1 i 1 In the equality holds if and only if x1 xn 8 A n . This inequality played an important role in number theory and has been generalized in several directions 1 3-6 . One of its generalizations states the following. Theorem 5 Corollary . Let X be a real or complex normed space with dual X and suppose p q 1 and 1 p 1 q 1. If 8 A 0 x e X and f e X then A p-1 18 - f x p Ap-1 x p j fq 8p. In the equality holds if and only if f x II f HUxh and xh 8 f q-1 A II f q . In this paper we give a new interpretation of the inequality and consider whether the coefficients Ap-1 and A A II f q p-1 are best possible. For this purpose we divide Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006 Article ID 21540 Pages 1-8 DOI JIA 2006 21540 2 A reconsideration of Hua s inequality. Part II both sides of by A A II f q p 1sp and then replace x s by x. Thus we obtain a replica of Theorem . Theorem . Let X be a real or complex normed space with dual X and suppose p q 1 and 1 p 1 q 1. If A 0 x e X and f e X then A A l q p 111 - f x p A IIf q p-1HxHp 1. In the equality holds if and only if f x II f II x and x II f q-1 A II f q . Clearly Theorems and are equivalent. So we turn our attention to Theorem .
