Research in the Mathematical Sciences ( IF 1.2 ) Pub Date : 2021-01-29 , DOI: 10.1007/s40687-021-00248-5 J. E. Pascoe
We classify functions \(f:(a,b)\rightarrow \mathbb {R}\) which satisfy the inequality
$$\begin{aligned} {\text {tr}}f(A)+f(C)\ge {\text {tr}}f(B)+f(D) \end{aligned}$$when \(A\le B\le C\) are self-adjoint matrices, \(D= A+C-B\), the so-called trace minmax functions. (Here \(A\le B\) 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
$$\begin{aligned} \det g(A) \det g(C)\le \det g(B) \det g(D) \end{aligned}$$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.
中文翻译：
跟踪minmax函数和激进的Laguerre–Pólya类
我们对满足不等式的函数\（f：（a，b）\ rightarrow \ mathbb {R} \）进行分类
$$ \ begin {aligned} {\ text {tr}} f（A）+ f（C）\ ge {\ text {tr}} f（B）+ f（D）\ end {aligned} $$当\（A \ le B \ le C \）是自伴矩阵\（D = A + CB \）时，所谓的迹线minmax函数。（如果\（BA \）是正半定的，则\（A \ le B \），并且f通过函数演算求值。）当且仅当其导数解析地继续到函数的自映射时，函数才是迹线minmax。上半平面。迹线minmax函数\（g = e ^ {-f} \）的负指数满足不等式
$$ \ begin {aligned} \ det g（A）\ det g（C）\ le \ det g（B）\ det g（D）\ end {aligned} $$对于上面的A， B， C， D。我们称这样的函数为等参式。我们证明，行列式等参函数在Laguerre-Pólya类的“自由基”中。我们推导了此类函数的完整表示形式，该表达式实质上是Laguerre–Pólya类中函数的Hadamard分解的连续版本。我们应用我们的结果给出黎曼假设的一些等效公式。
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