Trace minmax functions and the radical Laguerre-Pólya class

Abstract

We classify functions $f:(a,b)\to \mathbb{R}$ which satisfy the inequality $$\mathrm{tr}f(A)+f(C)\ge \mathrm{tr}f(B)+f(D)$$ when $A\le B\le C$ are self-adjoint matrices, $D=A+C-B$, the so-called trace minmax functions. (Here $A\le B$ if $B-A$ is positive semidefinite, and $f$ is evaluated via the functional calculus.) A function is trace minmax if and only if its derivative analytically continues to a self map of the upper half plane. The negative exponential of a trace minmax function $g=e^{-f}$ satisfies the inequality $$\det g(A)\det g(C)\le \det g(B)\det g(D)$$ for $A,B,C,D$ as above. We call such functions determinant isoperimetric. We show that determinant isoperimetric functions are in the "radical" of the the Laguerre-Pólya class. We derive an integral representation for such functions which is essentially a continuous version of the Hadamard factorization for functions in the the Laguerre-Pólya class. We apply our results to give some equivalent formulations of the Riemann hypothesis.

16 pages