Trace minmax functions and the radical Laguerre–Pólya class

Abstract

We classify functions $f:(a,b)\rightarrow \mathbb {R}$ which satisfy the inequality

$$\begin{aligned} {\text {tr}}f(A)+f(C)\ge {\text {tr}}f(B)+f(D) \end{aligned}$$

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

$$\begin{aligned} \det g(A) \det g(C)\le \det g(B) \det g(D) \end{aligned}$$

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 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 Laguerre–Pólya class. We apply our results to give some equivalent formulations of the Riemann hypothesis.