A H\"older-type inequality on a regular rooted tree

TL;DR

This paper proves a H"older-type inequality for functions on finite regular rooted trees, linking interaction energies summed over automorphisms to products of level-wise sums, using induction and automorphism group analysis.

Contribution

It introduces a novel inequality relating interaction energies on trees to level-wise sums, extending classical inequalities to tree structures.

Findings

Established a new inequality involving functions on regular rooted trees.

Connected the inequality to automorphism groups of the tree.

Used induction and H"older's inequality in the proof.

Abstract

We establish an inequality which involves a non-negative function defined on the vertices of a finite $m$-ary regular rooted tree. The inequality may be thought of as relating an interaction energy defined on the free vertices of the tree summed over automorphisms of the tree, to a product of sums of powers of the function over vertices at certain levels of the tree. Conjugate powers arise naturally in the inequality, indeed, H\"{o}lder's inequality is a key tool in the proof which uses induction on subgroups of the automorphism group of the tree.