Trace minmax functions and the radical Laguerre-Polya 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 $Aleq Bleq C$ are self-adjoint matrices, $D= A+C-B$, the so-called trace minmax functions. (Here $Aleq B$ 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-Polya 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-Polya class. We apply our results to give some equivalent formulations of the Riemann hypothesis.