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- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature 2021
- eISSN
- 2197-9847
- DOI
- 10.1007/s40687-021-00248-5
- Publisher site
- See Article on Publisher Site
Abstract
We classify functions f:(a,b)→R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f:(a,b)\rightarrow \mathbb {R}$$\end{document} which satisfy the inequality trf(A)+f(C)≥trf(B)+f(D)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} {\text {tr}}f(A)+f(C)\ge {\text {tr}}f(B)+f(D) \end{aligned}$$\end{document}when A≤B≤C\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A\le B\le C$$\end{document} are self-adjoint matrices, D=A+C-B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D= A+C-B$$\end{document}, the so-called trace minmax functions. (Here A≤B\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A\le B$$\end{document} if B-A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B-A$$\end{document} 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\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g=e^{-f}$$\end{document} satisfies the inequality detg(A)detg(C)≤detg(B)detg(D)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \det g(A) \det g(C)\le \det g(B) \det g(D) \end{aligned}$$\end{document}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.
Journal
Research in the Mathematical Sciences – Springer Journals
Published: Jan 29, 2021