Trace minmax functions and the radical Laguerre-Pólya class

Abstract

We classify functions f : (a, b) → R which satisfy the inequality tr f (A) + f (C) ≥ tr f (B) + f (D) when A ≤ B ≤ C are self-adjoint matrices, D = A + C − B, the so-called trace minmax functions. (Here A ≤ 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) ≤ 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.