A classical result of Aubin states that the constant in the Moser-Trudinger-Onofri inequality on can be improved for functions with zero first-order moments of the area element. We generalize it to the higher-order moments case. These new inequalities bear similarity to a sequence of Lebedev-Milin-type inequalities on coming from the work of Grenander-Szego on Toeplitz determinants (as pointed out by Widom). We also discuss the related sharp inequality by a perturbation method . © 2020 Wiley Periodicals LLC.

中文翻译：

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约束下改进的 Moser-Trudinger-Onofri 不等式

Aubin 的经典结果表明，对于面积元素的一阶矩为零的函数，可以改进Moser-Trudinger-Onofri 不等式中的常数。我们将其推广到高阶矩的情况。这些新的不等式与一系列 Lebedev-Milin 类型的不等式相似，这些不等式来自 Grenander-Szego 关于 Toeplitz 行列式的工作（如 Widom 所指出的）。我们还通过微扰方法讨论了相关的尖锐不等式。© 2020 威利期刊有限责任公司。