Sharpening H\"older's inequality

TL;DR

This paper introduces a new family of sharp inequalities that strengthen H"older's inequality, providing an analog of the Pythagorean theorem for L^p spaces using Bellman functions.

Contribution

It develops a novel approach to strengthen H"older's inequality by employing Bellman functions, offering a new perspective on inequalities in L^p spaces.

Findings

Established a new family of sharp inequalities for H"older's inequality.

Provided an analog of the Pythagorean theorem for L^p spaces.

Utilized Bellman functions of four variables in the proofs.

Abstract

We strengthen H\"older's inequality. The new family of sharp inequalities we obtain might be thought of as an analog of Pythagorean theorem for the $L^p$ spaces. Our reasonings rely upon Bellman functions of four variables.