Majorization inequalities via convex functions

IF 0.7 4区 数学 Q2 Mathematics
M. Kian, M. Sababheh
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引用次数: 0

Abstract

Convex functions have been well studied in the literature for scalars and matrices. However, other types of convex functions have not received the same attention given to the usual convex functions. The main goal of this article is to present matrix inequalities for many types of convex functions, including log-convex, harmonically convex, geometrically convex, and others. The results extend many known results in the literature in this direction. For example, it is shown that if $A,B$ are positive definite matrices and $f$ is a continuous $\sigma\tau$-convex function on an interval containing the spectra of $A,B$, then\begin{align*}\lambda^\downarrow (f(A\sigma B))\prec_w\lambda^\downarrow \left(f(A)\tau f(B)\right),\end{align*}for the matrix means $\sigma,\tau\in\{\nabla_{\alpha},!_{\alpha}\}$ and $\alpha\in[0,1]$. Further, if $\sigma=\sharp_{\alpha}$, then\begin{align*} \lambda^\downarrow \left(f\left(e^{A\nabla_{\alpha}B}\right)\right)\prec_w\lambda^\downarrow \left(f(e^A)\tau f(e^B))\right).\end{align*}Similar inequalities will be presented for two-variable functions too.
基于凸函数的优化不等式
凸函数在标量和矩阵的文献中得到了很好的研究。然而,其他类型的凸函数并没有像通常的凸函数那样受到重视。本文的主要目标是介绍多种凸函数的矩阵不等式,包括对数凸、调和凸、几何凸等。这些结果在这个方向上扩展了文献中许多已知的结果。例如,如果$A,B$是正定矩阵,$f$是包含$A,B$谱的区间上的连续$\sigma\tau$ -凸函数,则对于矩阵\begin{align*}\lambda^\downarrow (f(A\sigma B))\prec_w\lambda^\downarrow \left(f(A)\tau f(B)\right),\end{align*}表示$\sigma,\tau\in\{\nabla_{\alpha},!_{\alpha}\}$和$\alpha\in[0,1]$。进一步,如果$\sigma=\sharp_{\alpha}$,那么\begin{align*} \lambda^\downarrow \left(f\left(e^{A\nabla_{\alpha}B}\right)\right)\prec_w\lambda^\downarrow \left(f(e^A)\tau f(e^B))\right).\end{align*}对于两变量函数也会出现类似的不等式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.20
自引率
14.30%
发文量
45
审稿时长
6-12 weeks
期刊介绍: The journal is essentially unlimited by size. Therefore, we have no restrictions on length of articles. Articles are submitted electronically. Refereeing of articles is conventional and of high standards. Posting of articles is immediate following acceptance, processing and final production approval.
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