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

By Sever S. Dragomir

The aim of this ebook is to offer a accomplished creation to a number of inequalities in internal Product areas that experience very important functions in numerous subject matters of latest arithmetic equivalent to: Linear Operators concept, Partial Differential Equations, Non-linear research, Approximation concept, Optimisation concept, Numerical research, chance conception, facts and different fields.

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KUREPA, On the Buniakowsky-Cauchy-Schwarz inequality, Glasnik Matematˇcki, 1(21) (1966), 146-158. 1. Introduction Let H be a linear space over the real or complex number field K. The functional ·, · : H × H → K is called an inner product on H if it satisfies the conditions (i) x, x ≥ 0 for any x ∈ H and x, x = 0 iff x = 0; (ii) αx + βy, z = α x, z + β y, z for any α, β ∈ K and x, y, z ∈ H; (iii) y, x = x, y for any x, y ∈ H. 1) for any x, y ∈ H. , there exists a nonzero constant α ∈ K so that x = αy.

10) is obvious. Corollary 4 (Dragomir, 1985). 14) x y ≥ 2 | x, e e, y | . Remark 11. Assume that A : H → H is a bounded linear operator on H. 15) Ay ≥ | x, Ay − x, e e, Ay | + | x, e e, Ay | ≥ | x, Ay | for any y ∈ H. 16) Ay = sup {| x, Ay − x, e e, Ay | + | x, e e, Ay |} x =1 for any y ∈ H. 17) A = sup {| x, Ay − x, e e, Ay | + | x, e e, Ay |} y =1, x =1 for any e ∈ H, e = 1, a representation that has been obtained in [15, Eq. 9]. 2. INEQUALITIES RELATED TO SCHWARZ’S ONE 41 Remark 12. Let (H; ·, · ) be a Hilbert space.

47) 1 2 | v, t |2 + | w, t |2 + | v, t |2 − | w, t |2 2 + 4 (Re v, t Re w, t + Im v, t Im w, t )2 1 ≤ t 2 ≤ v 2 2 v + w 2 + w 2 t 2 2 + v 2 − w 2 2 1 2 1 2 2 + 4 Re (v, w) , for all v, w, t ∈ H. Proof. 48) |(cos ϕ · v + sin ϕ · w, z)|2 ≤ cos ϕ · v + sin ϕ · w for any ϕ ∈ [0, 2π] . 4. REFINEMENTS OF BUZANO’S AND KUREPA’S INEQUALITIES 51 Since I (ϕ) := cos ϕ · v + sin ϕ · w = cos2 ϕ v 2 2 + 2 Re (v, w) sin ϕ cos ϕ + w 2 sin2 ϕ, hence, as in Theorem 16, 1 2 sup I (ϕ) = ϕ∈[0,2π] v 2 + w 2 + v 2 − w 2 2 + 4 Re2 (v, w) 1 2 .