On an inequality of Osgood, Phillips and Sarnak

Abstract:

In their work [4, 5] on isospectral families for the Laplacian Osgood, Phillips and Sarnak needed and proved the following inequality: For any real-valued function $\phi$ belonging to the Sobolev space ${H_{1/2}}( = {W^{1/2,2}})$ of the unit circle and satisfying the side condition $\int {{e^\phi }{e^{i\theta }}d\theta = 0}$ one has \[ \log \frac {1} {{2\pi }}\int {{e^\phi }d\theta - \frac {1} {{2\pi }}\int \phi d\theta \leq \frac {1} {2}\sum \limits _{k = 1}^\infty {k|\hat \phi } (k){|^2}} \] where $\hat \phi (k)$ is the $k$th Fourier coefficient of $\phi$. Without the side condition the factor $\frac {1} {2}$ does not appear on the right side and the inequality is the first Lebedev-Milin inequality [1, §5.1]. The purpose of this note is to show that these are the first two of a series of inequalities which follow quickly from some theorems of G. Szegö on Toeplitz determinants.