TAILIEUCHUNG - Báo cáo " The parameter-dependent cyclic inequality "

In this paper we will construct a parameter-dependent cyclic inequality that can be used to prove a lot of hard and interesting inequalities. 1. Introduction The cyclic inequality is a type of inequality that may be right in just some particular cases but not in genenal. In this paper, we propose one type of parameter-dependent cyclic inequality from a special inequality. Thanks to this inequality, we can obtain many inequalities by choosing α and n. Note that it can be proved by some ways in particular case. However in order to prove it in general case, we have to. | VNU Journal of Science Mathematics - Physics 23 2007 155-158 The parameter-dependent cyclic inequality Nguyen Vu Luong Department of Mathematics Mechanics Informatics College of Science VNU 334 Nguyen Trai Hanoi Vietnam Received 15 November 2006 received in revised form 12 September 2007 Abstract. In this paper we will construct a parameter-dependent cyclic inequality that can be used to prove a lot of hard and interesting inequalities. 1. Introduction The cyclic inequality is a type of inequality that may be right in just some particular cases but not in genenal. In this paper we propose one type of parameter-dependent cyclic inequality from a special inequality. Thanks to this inequality we can obtain many inequalities by choosing a and n. Note that it can be proved by some ways in particular case. However in order to prove it in general case we have to use the method that is mentioned in the paper. 2. The general case Denote R x G Rx 0 . Lemma . Assume that xi G R i 1 n we have Proof. We have IS xixj -1 i j n IS xixj-1 i j n X2 xj 2 1 i j n . . n n -1 . n n - 1 Since 1 2 n 1 ----- hence the number of terms of 2J 1 i j n xixj is - It follows 2 E xx - z x2 xj n 1 Ex2 . 1 i j n 1 i j n i 1 E-mail luongnv@ 155 156 Nguyen Vu Luong VNU Journal of Science Mathematics - Physics 23 2007 155-158 Adding both sides of the above inequality by was to be proved. The proof of Lemma is complete. Theorem . Assume that Xi i 1 n following inequality 2 n 1 Ei i j n XiXj we obtain the inequality as n 3 are positive number. Then there holds the X -------------- T Xi T a X2 T-----T CnXk 1 Xn X2 ------7---------------7--T T X2 a X3 T---T CnXk 2 2n ĨT 1-1 Xn a Xi T----T CnXk 2 a n 1 Where Cn n 2 k 2 and a is an arbitrary real number satisfies a 2. Proof. First for the sake convinience we set p X1 X2 P --------7------------------T T--------7-----------------r T T Xi T a X2 T------T CnXk 1 X2 T a X3 T------T CnXk 2 Xn 2n T T ------7---------------7 -------7---77 . .

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