Trace minmax functions and the radical Laguerre–Pólya class

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 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.