# A Sharp Inequality of Hardy-Littlewood Type Via Derivatives

## TL;DR

This paper investigates a generalized Hardy-Littlewood inequality for holomorphic functions, analyzing the behavior of associated norms via derivatives, and establishes the inequality for specific function classes with numerical support.

## Contribution

It introduces a derivative-based approach to verify a generalized Carleman's inequality for holomorphic functions, including cases involving linear combinations of kernels.

## Key findings

- Norms $\
- Derivative of norms $\

## Abstract

In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\|f\|_{A_\alpha^{2\alpha}}\leq\|f\|_{H^2}$, where $f$ is a holomorphic function and $\alpha>1$. If the norms $\|f\|_{A_\alpha^{2\alpha}}$ are decreasing in $\alpha$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\|f\|_{A_\alpha^{2\alpha}}$ 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.