Edwin Beckenbach

On the Positivity of Circulant and Skew-Circulant Determinants

For a given vector A := (a0,a1,...,an−1) in ℝn, simple necessary and sufficient conditions on a0,a1,...,an−1 are established for the determinant of the circulant matrix of A to be positive, or negative, or zero. There is a striking difference between the conditions for n odd and the conditions for n even. The determinant of the skew-circulant matrix of A is similarly discussed.

The Classical Inequalities

Arithmetic mean-Geometric mean inequality says that for any n non-negative real numbers a 1 , ...a n we have: a 1 + a 2 + ... + a n n ≥ n √ a 1 a 2 ...a n and equality holds ⇔ a 1 = a 2 = ... = a n. Define the Harmonic mean by H = n 1/a 1 + 1/a 2 + ... + 1/a n then it is not hard to check that HM ≤ GM ≤ AM. Theorem 1 (Power mean inequality) Let a 1 , ...a n be positive real numbers , and let α be real. Let M α (a 1 , ...a n) = (a α 1 + ... + a α n n) 1/α , α = 0 and M 0 = n √ a 1 ...a n. Then M α is an increasing function of α unless a 1 = a 2 = ... = a n (in which case M α is constant). Theorem 2 (Cauchy inequality) For arbitrary real numbers a 1 , a 2 , ...a n and b 1 , b 2 , ...b n we have (a 1 b 1 + ... + a n b n) 2 ≤ (a 2 1 + ... + a 2 n)(b 2 1 + ... + b 2 n) Furthermore, equality holds if and only if there are two numbers λ, µ not both zero, such that λa i = µb i for all i.

The Fundamental Inequalities and Related Matters

In this initial chapter, we shall present many of the fundamental results and techniques of the theory of inequalities. Some of the results are important in themselves, and some are required for use in subsequent chapters; others are included, as are multiple proofs, on the basis of their elegance and unusual flavor [1].

On the Positivity of Operators

In this chapter, we shall explore the following theme: “Given a set of functions {u} satisfying certain side conditions, and an operator L that can he applied to the functions of this set, determine when the inequality n n

Moment Spaces and Resonance Theorems

Lleft( u right) geqq 0

Note to the Reader

n n(1) n nimplies that u ≧ 0.” An operator will be said to be “positive” if this condition is satisfied, although it might be more reasonable to call the inverse operator L −1 positive. We shall focus our attention upon ordinary differential and partial differential operators.

An Introduction to Inequalities: Properties of Distance

As soon as we leave the well-traversed fields of real and complex numbers for the broader and relatively unexplored domains of hyper-complex numbers, we open the way for the introduction of many different types of ordering relationships. In this chapter, we shall discuss a variety of interesting inequalities centering about the theme of matrices. As we shall see, the basic concept of positive number can be extended to matrices in many different and significant ways.

An Introduction to Inequalities: Fundamentals

A central idea of analysis, which can be used to connect vast fields of study that at first glance may seem quite unrelated, can be expressed in the following simple form: n“An element of a linear space S can often be characterized most readily and revealingly in terms of its interaction with a suitably chosen set of elements in a dual space S′.”