Advances in Inequalities of the Schwarz, Triangle and by Sever S. Dragomir

By Sever S. Dragomir

Booklet by way of Dragomir, Sever S.

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53) n n n n n 1 2 2 p i ai x i p i ai x i ≤ pi |ai | + p i ai pi |xi |2 , 2 i=1 i=1 i=1 i=1 i=1 for any pi ≥ 0, ai , xi , bi ∈ C, i ∈ {1, . . , n} . Note that, if xi , i ∈ {1, . . 54) pi xi zi i=1 1 ≤ 2 n n i=1 n 2 pi x2i pi zi2 pi |zi | + i=1 , i=1 where zi ∈ C, i ∈ {1, . . , n} . In this way, Buzano’s result may be regarded as a generalisation of de Bruijn’s inequality. Similar comments obviously apply for integrals, but, for the sake of brevity we do not mention them here. 4. REFINEMENTS OF BUZANO’S AND KUREPA’S INEQUALITIES 53 The aim of the present section is to establish some related results as well as a refinement of Buzano’s inequality for real or complex inner product spaces.

40). 42) ≤ cos ϕ · x + sin ϕ · y 2 z 2 . 41) is proved. 41) is proved. Remark 14. 43) x, z 2 + y, z 2 1 ≤ x 2+ y 2+ x 2 ≤ x 2+ y 2 z 2. 2 − y 2 2 + 4 x, y 2 1 2 z 2 50 2. SCHWARZ RELATED INEQUALITIES Remark 15. If H is a real space, ·, · the real inner product, HC its complexification and ·, · C the corresponding complexification for ·, · , then for x, y ∈ H and w := x + iy ∈ HC and for e ∈ H we have Im x, e w 2 C 2 = x 2 + y C = Im y, e | w, w¯ C | = , C = 0, x 2 − y 2 2 + 4 x, y 2 , where w¯ = x − iy ∈ HC .

Let us consider the mapping py : H × H → K, py (x, z) = x, z 2 y − x, y y, z for each y ∈ H\ {0} . 2). Remark 9. 2. INEQUALITIES RELATED TO SCHWARZ’S ONE 39 for every x, y, z ∈ H. 5) y 2 − | x ± y, y |2 ≤ x 2 y 2 − | x, y |2 for every x, y ∈ H. 5) have been obtained in [15]. 6) sup x + λy 2 y 2 − | x + λy, y |2 = x 2 y 2 − | x, y |2 λ∈K for each x, y ∈ H. 2]): Corollary 3 (Dragomir, 1985). 7) y, z + y z 2 z, x + z x 2 x, y y, z z, x . x 2 y 2 z 2 ≤ 1+2 Proof. By the modulus properties we obviously have x, z y 2 − x, y y, z ≥ | x, z | y 2 − | x, y | | y, z | .

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