Trace minmax functions and the radical Laguerre–Pólya class

We classify functions $$f:(a,b)\rightarrow \mathbb {R}$$ which satisfy the inequality $$\begin{aligned} {\text {tr}}f(A)+f(C)\ge {tr}}f(B)+f(D) \end{aligned}$$ when $$A\le B\le C$$ are self-adjoint matrices, $$D= A+C-B$$ , so-called trace minmax functions. (Here B$$ if $$B-A$$ is positive semidefinite, and f evaluated via functional calculus.) A function only its derivative analytically continues to a self-map of upper half plane. The negative exponential $$g=e^{-f}$$ satisfies \det g(A) g(C)\le g(B) g(D) for A, B, C, D as above. call such determinant isoperimetric. show that isoperimetric in “radical” Laguerre–Pólya class. derive an integral representation essentially continuous version Hadamard factorization apply our results give some equivalent formulations Riemann hypothesis.