Trace minmax functions and the radical Laguerre-Pólya class

We classify functions $f:(a,b)\rightarrow \mathbb{R}$ which satisfy the inequality $\operatorname{tr} f(A)+f(C)\geq \operatorname{tr} f(B)+f(D)$ when $A\leq B\leq C$ are self-adjoint matrices, $D= A+C-B$, the so-called trace minmax functions. (Here $A\leq B$ 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 $\det g(A) \det g(C)\leq \det g(B) \det g(D)$ 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.

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