An Improvement of Hardy’s Inequality for Hilbert Space Operators

Abstract

In this paper, using the notion of superquadratic functions and operator valued mappings, we give an improvement of the Hardy inequality for Hilbert space operators. Then we apply our results to the classical Hardy inequality. In particular, we obtain an estimation for the positive expression $$\begin{aligned} \left( \frac{p}{p-1}\right) ^p\int _{0}^{\infty }f(x)^pdx - \int _{0}^{\infty }\left( \frac{1}{x}\int _{0}^{x}f(t)dt\right) ^pdx, \end{aligned}$$ where $$p\ge 1$$ and f is a positive p-integrable function on $$(0,\infty )$$ .