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

By Sever S. Dragomir

The aim of this ebook is to provide a finished advent to numerous inequalities in internal Product areas that experience vital functions in numerous issues of up to date arithmetic equivalent to: Linear Operators concept, Partial Differential Equations, Non-linear research, Approximation concept, Optimisation concept, Numerical research, likelihood conception, data and different fields.

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Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces

The aim of this publication is to provide a accomplished creation to a number of inequalities in internal Product areas that experience very important functions in quite a few issues of up to date arithmetic similar to: Linear Operators conception, Partial Differential Equations, Non-linear research, Approximation conception, Optimisation concept, Numerical research, likelihood concept, information and different fields.

Extra info for Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces

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33) x y ≥ | x, ei ei , y | x, ei ei , y + i∈F i∈F ≥2 x, ei ei , y , i∈F for any nonempty finite part of I. 3. 1. Kurepa’s Inequality. G. de Bruijn proved the following refinement of the celebrated Cauchy-Bunyakovsky-Schwarz (CBS) inequality for a sequence of real numbers and the second of complex numbers, see [2] or [9, p. 48]: Theorem 14 (de Bruijn, 1960). Let (a1 , . . , an ) be an n−tuple of real numbers and (z1 , . . , zn ) an n−tuple of complex numbers. 34) ak zk k=1 1 ≤ 2 n n k=1 n k=1 zk2 |zk | + k=1 n k=1 |zk |2 a2k · ≤ n 2 a2k .

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. An improvement of Kurepa’s inequality for the complexification of a real inner product and the corresponding applications for discrete and integral inequalities are also provided.

Some Refinements. The following result holds [15, Theorem 1] (see also [18, Theorem 2]). Theorem 10 (Dragomir, 1985). Let (H, ·, · ) be a real or complex inner product space. 2) x 2 2 y − | x, y |2 y 2 z 2 − | y, z |2 ≥ x, z 2 y − x, y y, z 2 for any x, y, z ∈ H. Proof. We follow the proof in [15]. 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.

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