A Hardy-Littlewood Integral Inequality on Finite Intervals with a Concave Weight

TL;DR

This paper establishes a Hardy-Littlewood integral inequality for functions on finite intervals with a concave weight, providing bounds relating the weighted norms of a function and its derivatives.

Contribution

It introduces a new inequality involving weighted norms with concave weights, extending classical Hardy-Littlewood inequalities to finite intervals.

Findings

The inequality bounds the weighted L^2 norm of the derivative by the product of weighted norms of the function and its second derivative.

The result applies to functions with specific boundary conditions on finite intervals.

The inequality generalizes existing results to include concave weight functions.

Abstract

A Hardy-Littlewood integral inequality on finite intervals with a concave weight is established. Given a function f on an interval [a,b], it is shown that the square of the weighted L^2 norm of its derivative f' is bounded by the product of the weighted L^2 norm of f and that of the second derivative f''.