Mixed-norm estimates and symmetric geometric means

TL;DR

This paper introduces new mixed-norm inequalities using symmetric geometric means, unifies existing estimates, simplifies proofs, and derives a novel inequality combining previous results.

Contribution

It presents a general framework for mixed-norm estimates via symmetric geometric means, unifying and extending prior inequalities with simpler proofs and new results.

Findings

Unified existing mixed-norm estimates as special cases

Simplified proofs for various inequalities

Derived a new inequality combining previous results

Abstract

The mixed-norm versions of the H\"older and Minkowski integral inequalities are used to produce new, general estimates involving symmetric geometric means of mixed norms. Various existing mixed-norm estimates are shown to be simple special cases of these new results. Examples are also given of applying mixed-norm H\"older and Minkowski to other estimates, providing much easier proofs. Finally, the effectiveness of this technique is demonstrated by deriving a new inequality which combines features from two separate previous results.