Hardy's inequality related to a Bernoulli equation

Abstract

The weighted Hardy's inequality is known asbegin{displaymath}int_a^b |u(x)|^{p}r(x)dx leq Cint_a^b|u'(x)|^{p}s(x)dxend{displaymath}where -infty leq a leq b leq infty and 1<p< infty. Thepurpose of this article is to provide a useful formulato express the weight r(x) in terms of s(x) or vice versaemploying a Bernoulli equation having the other weight ascoefficients.