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

By Sever S. Dragomir

The aim of this ebook is to provide a entire creation to numerous inequalities in internal Product areas that experience very important functions in numerous issues of latest arithmetic comparable to: Linear Operators idea, Partial Differential Equations, Non-linear research, Approximation concept, Optimisation idea, Numerical research, likelihood concept, facts and different fields.

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35). 44) 2 w 2 C , Corollary 7. Let x, y, z be as in Theorem 16. 45) 1 2 ≤ z · max { x , y } . Remark 16. 46) x, z 2 + y, z 1 2 2 ≤ z · max { x , y } . 3. A Related Result. Utilising Lemma 2, we may state and prove the following result as well. Theorem 17 (Dragomir, 2004). Let (H, ·, · ) be a real or complex inner product 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.

Let (H; ·, · ) be a Hilbert space. Then for any continuous linear functional f : H → K, f = 0, there exists, by the Riesz representation theorem a unique vector e ∈ H\ {0} such that f (x) = x, e for x ∈ H and f = e . 18) f 2 ≥ 2 | x, e y, e | = 2 |f (x)| |f (y)| for any x ∈ E and y ∈ E ⊥ . 19) f 2 ≥2 f E · f E⊥ for any E a nonzero linear subspace of the Hilbert space H and a given functional f ∈ H ∗ \ {0} . 19) has been obtained in [15, Eq. 10]. 2. A Conditional Inequality. The following result providing a lower bound for the norm product under suitable conditions holds [19] (see also [18, Theorem 1]): Theorem 12 (Dragomir-S´andor, 1986).

8) x, y y, z z, x 3 x 2 y 2 z 2 ≤ x, y x y 2 + y, z y z 2 + z, x z x 2 for any x, y, z ∈ H\ {0} . 8). 1]): 40 2. SCHWARZ RELATED INEQUALITIES Theorem 11 (Dragomir, 1985). 9) y ≥ | x, y − x, e e, y | + | x, e e, y | ≥ | x, y | . x Proof. We follow the proof in [15]. 10) x 2 − | x, e |2 y 2 − | y, e |2 ≥ | x, y − x, e e, y |2 . 12) y − | x, e e, y |)2 ( x ≥ x 2 − | x, e |2 y 2 − | y, e |2 for any x, y, e ∈ H with e = 1. 10). 10) is obvious. Corollary 4 (Dragomir, 1985). 14) x y ≥ 2 | x, e e, y | .

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