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By S.S.Dragomir

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Xn ) , y ¯= (y1 , . . , yn ) and ¯ z = (z1 , . . , zn ) . If yk2 ≤ |xk zk | for any k ∈ {1, . . 31) then one has the inequality: 2 n |yk | k=1 n n ≤ |zk | . 32) 48 CHAPTER 2. REFINEMENTS OF THE (CBS) −INEQUALITY Proof. We will follow the proof in [2]. 32). 6]. ¯ = (b1 , . . , bn ) and ¯ Theorem 77 Let ¯ a = (a1 , . . , an ), b c = (c1 , . . , cn ) be sequences of real numbers such that (i) |bk | + |ck | = 0 (k ∈ {1, . . , n}) (ii) |ak | ≤ 2|bk ck | |bk |+|ck | for any k ∈ {1, . . , n} .

The second part goes likewise and we omit the details. Remark 85 The following particular inequalities provide refinement for the (CBS) −inequality [3, p. 60 – p. 61]. 1. Assume that ¯ a = (a1 , . . , an ), b = (b1 , . . 7. DE BRUIJN’S INEQUALITY 55 quences of real numbers. Then n 2 n n a2i b2i − ai bi i=1   [ ≥ max    i=1 i=1 [ n i=1 a2i n i=1 bi − n n a2i − ( i=1 n i=1 bi n i=1 n n i=1 n i=1 ai b i − b2i − ( n i=1 n i=1 ai 2 ai b i ] , 2 ai ) n i=1 n 2 i=1 bi ] ai 2 n i=1 bi )   2  .

N) , one has: n k=1 n k=1 |xk | n k=1 (|xk | 2 |xk yk | ≤ |xk | + |yk | n k=1 |yk | . 37) For two positive real numbers, let us recall the following means a+b 2 √ G (a, b) := ab A (a, b) := and H (a, b) := 1 a 2 + (the arithmetic mean) (the geometric mean) 1 b (the harmonic mean). ¯ = (b1 , . . , bn ) are sequences of real We remark that if ¯ a = (a1 , . . 38) bi . 39) and, by the (CBS) −inequality, n G (ai , bi ) ≤ G i=1 ai , i=1 i=1 The following similar result for harmonic means also holds [2, p.

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A Survey on Cauchy-Buniakowsky-Schwartz Type Discrete Inequalities by S.S.Dragomir


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