A Sharp Inequality on the Exponentiation of Functions on the Sphere

TL;DR

This paper introduces a new inequality on the sphere that generalizes classical inequalities like Lebedev-Milin and Moser-Trudinger, incorporating mass center deviation and improving existing bounds.

Contribution

It extends classical inequalities to the sphere, explicitly accounts for mass center shifts, and connects to Toeplitz determinant inequalities and Aubin's inequality.

Findings

Derived a new inequality on the sphere generalizing Lebedev-Milin

Incorporated mass center deviation into classical inequalities

Improved Onofri's inequality with explicit mass shift contribution

Abstract

In this paper we show a new inequality which generalizes to the unit sphere the Lebedev-Milin inequality of the exponentiation of functions on the unit circle. It may also be regarded as the counterpart on the sphere of the second inequality in the Szeg\"o limit theorem on the Toeplitz determinants on the circle. On the other hand, this inequality is also a variant of several classical inequalities of Moser-Trudinger type on the sphere. The inequality incorporates the deviation of the center of mass from the origin into the optimal inequality of Aubin for functions with mass centered at the origin, and improves Onofri's inequality with the contribution of the shifting of the mass center explicitly expressed.