OVIDIU T. POP. 1. Introduction. The inequality from Theorem 1 is called in the literature Bergström s inequality (see [2], [3], [4], [6]).

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1 Journal of Mathematical Inequalities Volume 3, Number 009, 37 4 ABOUT BERGSTRÖM S INEQUALITY OVIDIU T POP Communicated by P Bullen Abstract In this paper, we generalize identity 3, from where we obtain a rafinement of inequalities and Introduction The inequality from Theorem is called in the literature Bergström s inequality see [], [3], [4], [6] THEOREM If x k R and a k > 0, k {,,,n},then x a x a x n a n x x x n a a a n, x with equality if and only if x x n a a a n The generalizationof Bergström s inequality is contained in the following theorem see [5] THEOREM If x k R and a k > 0, k {,,,n},then x x x n x x x n a i x j a i x j max a a a n a a a n a i a j By particularizations in Theorem, in paper [5] the refinements are obtained of Cauchy-Schwarz s inequality For the complex numbers, it is well-known the identity: z a z a z z a a a z a z a a a a 3 Mathematics subject classification 000: 6D5 Keywords and phrases: Bergström s inequality, Lagrange s identity, multivariate inequalities refinements c D l,zagreb Paper JMI

2 38 OVIDIU T POP true for any z,z C and any a,a R\{0} such that a a 0see[6],page 35 In the following, we note N {,,} Main results In this section we start with the generalization of identity 3 THEOREM 3 If n N, n, z,z,,z n C and a,a,,a n R\{0},such that a a a n 0,then z z z n z z z n 4 a a a n a a a n a i z j a j z i a a a n Proof We consider z k x k iy k, x k,y k R, k {,,,n} Then the member from the left from the 4 is equivalent with x y x y x ny n x x x n y y y n a a a n a a a n a a a n a a a n a a 3 a 4 a n a a 3a 4 a n a a 3 a n a nx y a a 3a 4 a n a a 3 a 4 a n a a 3 a n a n x y a a a 3 a n a a a 3 a n a a a n a nx ny n a a a n x x x x 3 x x n x n x n y y y y 3 y y n y n y n a3 a 4 a n a x a x a y a y a a a n a a a n a a 4 a 5 a n a x 3 a 3 x a y 3 a 3 y a a a n an x n a n x n a n y n a n y n a3 a 4 a n a z a z a a a n a a a n a a 4 a 5 a n a z 3 a 3 z a a a n a n z n a n z n, from where, the relation 4 results

3 ABOUT BERGSTRÖM S INEQUALITY 39 COROLLARY If n N, n, x,x,,x n R and a,a,,a n R\{0},such that a a a n 0,then x a x a x n a n x x x n a a a n 5 a i x j a j x i a a a n Proof In the identity 4 we consider z k x k R, k {,,,n} COROLLARY If n N, x, x k,y k R, k {,,,n},then x x x n y y y n 6 x y x y x n y n x i y j x j y i Proof Changing a k by y k and x k by x k y k, k {,,,n} in identity 5, we obtain identity 6 REMARK The identity from Corollary is called Lagrange s identity COROLLARY 3 If z,z,,z n C and a,a,,a n 0,, then z z z n z z z n, 7 a a a n a a a n with equality if and only if a i z j a j z i for any i, j {,,,n} Proof The inequality 7 results immediately from Theorem 3 THEOREM 4 If n N, n, z,z,,z n C and a,a,,a n R\{0} such that a a a n > 0,then z a z a z n a n a a a n with equality if and only if z z z n 0 Proof It results from Theorem 3 COROLLARY 4 If n N, n and z,z,,z n C, then a i z j a j z i, 8 z z z n n z j z i 9

4 40 OVIDIU T POP Proof The inequality 9 results from Theorem 4 if we take a a a n where THEOREM 5 If n N, n, x,x,,x n R and a,a,,a n 0,, then x x x n x x x n 0 a a a n a a a n a i x j a j x i A k,l a a a n, A k,l max Proof We have that T n m m m a i x j a j x i a i a j a kx l a l x k, k < l n a k a l a k a l am x l a l x m x k a k x m a l a k a k x l a la k x m a l a k x m a l x k al a k and applying the inequality for n, we obtain that T a k a l n a k x l a l x k n a k x l a l x k a k a l a k a l m m a k a l a k a l Taking the inequality above into account, we have that a i x j a j x i a a a n a k x l a l x k n am x l a l x m a a a n a k a l x k a k x m m a l x k a a a n a a a n a i x j a j x i a k x l a l x k a k a l n m a i x j a j x i a a a n, a k x l a l x k a k a l a k a l from where and taking identity 5 into account, the inequality 0 results

5 ABOUT BERGSTRÖM S INEQUALITY 4 REMARK From Theorem 5 the inequalities from and results THEOREM 6 If n N, n, x i,y i R, i {,,,3},then n x i n y i n y i if y i 0 for any i {,,,n}, n x i n y i n x i n x i y i x k y l x l y k y k y l x i y j x j y i n x i y i x p y q x q y p x p x q x i y j x j y i i, j {p,q} if x i 0 for any i {,,,}, wherek,l, p,q {,,,n}, k l, p q such that x max i y j x j y i x i y j x j y i is obtained for i k and j l, max y i y j x i is obtained x j for i p and j q Proof In Theorem 5 we change a i by y i and x i with x i y i, i {,,,n} and we obtain the inequality, respectively x i by y i, i {,,,n} in inequality and then we obtain inequality COROLLARY 5 If n N, n, x i,y i R\{0}, i {,,,n}, then the inequalities n x i n y i n x i y i n y i x i y j x j y i y i y j 3 and n x i n y i hold for any i, j {,,,n} n x i y i n xi y j x j y i x i x i y j 4 Proof It results from Theorem 6 REMARK 3 Using the inequalities from Corollary 5, we obtain some inequalities from paper [5]

6 4 OVIDIU T POP REFERENCES [] G HARDY, J E LITTLEWOOD AND G PÓLYA, Inequalities, Cambridge, 934 [] E F BECHENCBACH AND R BELLMAN, Inequalities, Springer, Berlin, Göttingen and Heidelberg, 96 [3] R BELLMAN, Notes on Matrix Theory - IV An Inequality Due to Bergström, Amer Math Montly, Vol 6 955, 7 73 [4] H BERGSTRÖM, A triangle inequality for matrices, Den Elfte Skandinaviske Matematikerkongress, 949, Trodheim, Johan Grundt Tanums Forlag, Oslo, 95, [5] D MĂRGHIDANU, J L DÍAZ-BARRERO AND S RĂDULESCU, New refinements of some classical inequalities, to appear in Math Ineq& Appl [6] D S MITRINOVIĆ, Analytic Inequalities, Springer-Verlag, Berlin, 970 Received October, 008 Ovidiu T Pop National College Mihai Eminescu 5 Mihai Eminescu Street Satu Mare Romania ovidiutiberiu@yahoocom Journal of Mathematical Inequalities wwwele-mathcom jmi@ele-mathcom

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