# Discrete Inequalities of Wirtinger’s Type

• Igor Ž. Milovanović
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 430)

## Abstract

Various discrete versions of Wirtinger’s type inequalities are considered. A short account on the first results in this field given by Fan, Taussky and Todd [10] as well as some generalisations of these discrete inequalities are done. Also, a general method for finding the best possible constants A n and B n in inequalities of the form
$$A_{n}\sum_{k=1}^{n}p_kx_{k}^{2}\leq\sum_{k=0}^{n}r_{k}(x_{k}-x_{k+1})^{2}\leq B_{n}\sum_{k=1}^{n}p_kx_{k}^{2}$$
where p = (p k ) and r = (r k ) are given weight sequences and x = (x k ) is an arbitrary sequence of the real numbers, is presented. Two types of problems are investigated and several corollaries of the basic results are obtained. Further generalisations of discrete inequalities of Wirtinger’s type for higher differences are also treated.

## Key words and phrases

Discrete inequalities Difference Eigenvalues and eigenvectors Best constants Orthogonal polynomials

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## Authors and Affiliations

• 1
• Igor Ž. Milovanović
• 1
1. 1.Department of MathematicsFaculty of Electronic EngineeringNišYugoslavia