Sums of Powers and Majorization

Symmetric polynomials, p-norm inequalities, and certain functionals related to majorization

We study a variant of the majorization relation. In particular we consider inequalities involving some Schur-concave symmetric polynomials related to the multinomial expansion. We also discuss how these topics were motivated by conjectures on sharp Lp inequalities between complex exponential sums conjectured by Hardy and Littlewood (still open problems), and why the usual majorization relation does not hold in that context.

Power majorization between the roots of two polynomials

It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization relation. In particular, a numerical observation by Klemeš is confirmed.

A Note on Sums of Powers

We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.

关于幂和商的上确界

On the supremum of a quotient of power sums

本文研究了幂和商函数的上确界问题，并将其应用于数学金融中的价格影响模型。

A Short Proof of a Known Relation for Consecutive Power Sums

We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers.

On sums of powers of cosecs

The finite sums of powers of cosecs occur in numerous situations, both physical and mathematical, examples being the Casimir effect, Renyi entropy, Verlinde's formula and Dedekind sums. I here present some further discussion which consists mainly of a reprise of early work by H.M.Jeffery in 1862-64 which has fallen by the wayside and whose results are being reproduced up to the present day. The motivation is partly historical justice and partly that, because of the continuing appearance of the sums, his particular methods deserve re--exposure. For example, simple trigonometric generating funct…