A Sharp Inequality of Hardy-Littlewood Type Via Derivatives

Abstract

In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\Vert f{\Vert }_{A_{\alpha }^{2\alpha }}\le \Vert f{\Vert }_{H^2}$, where $f$ is a holomorphic function and $\alpha >1$. If the norms $\Vert f{\Vert }_{A_{\alpha }^{2\alpha }}$ are decreasing in $\alpha$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\Vert f{\Vert }_{A_{\alpha }^{2\alpha }}$ in $\alpha$ and give sufficient conditions for this derivative to be non-positive. As an application, we prove the inequality for linear combinations of two reproducing kernels. Some numerical evidences are also provided.

23 pages, 4 figures

References in corpus (1)

• An inequality of Hardy--Littlewood type for Dirichlet polynomials