Inequalities involving polynomials and quasimodular forms

TL;DR

This paper investigates inequalities involving polynomials and quasimodular forms, focusing on monotonicity properties and applications to constructing positive forms and proving key inequalities in lattice packing problems.

Contribution

It introduces new monotonicity results for functions involving quasimodular forms and constructs infinitely many positive quasimodular forms of higher level.

Findings

Established monotonicity of specific functions involving quasimodular forms.

Constructed infinitely many positive quasimodular forms of level greater than 1.

Provided alternative proofs for key inequalities in lattice packing theory.

Abstract

In this paper, we study inequalities involving polynomials and quasimodular forms. More precisely, we focus on the monotonicity of the functions of the form $t \mapsto t^m F(it)$ where $F$ is a quasimodular form and $m > 0$. As an application, we construct infinitely many positive quasimodular forms of level $> 1$. We also give alternative proofs of modular form inequalities used in the proof of optimality of Leech lattice packing and universal optimality of the lattice by Cohn, Kumar, Miller, Radchenko, and Viazovska.