On Generalized Sharpness of Some Hardy-Type Inequalities

The current status concerning Hardy-type inequalities with sharp constants is presented in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure dx with the Haar measure \(dx/x.\) We also present some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also where the involved function spaces are optimal. As applications, a number of both well-known and new Hardy-type inequalities are pointed out. These results are used to present some new information concerning sharpness in the relation between different quasi-norms in Lorentz spaces.