Motivated by Horn's log-majorization (singular value) inequality $s\left(AB\right)\underset{log}{\prec }s\left(A\right)\ast s\left(B\right)$ and the related weak-majorization inequality $s\left(AB\right)\underset{w}{\prec }s\left(A\right)\ast s\left(B\right)$ for square complex matrices, we consider their Hermitian analogs $\lambda \left(\sqrt{A}B\sqrt{A}\right)\underset{log}{\prec }\lambda \left(A\right)\ast \lambda \left(B\right)$ for positive semidefinite matrices and $\lambda (|A\circ B|)\underset{w}{\prec }\lambda \left(|A|\right)\ast \lambda \left(|B|\right)$ for general (Hermitian) matrices, where $A\circ B$ denotes the Jordan product of A and B and $\ast$ denotes the componentwise product in ${\mathcal{R}}^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $\lambda (P_{\sqrt{a}}\left(b\right))\underset{log}{\prec }\lambda \left(a\right)\ast \lambda \left(b\right)$ for $a,b\ge 0$ and $\lambda (|a\circ b|)\underset{w}{\prec }\lambda \left(|a|\right)\ast \lambda \left(|b|\right)$ for all a and b, where $P_u$ and $\lambda \left(u\right)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form $\lambda (|A\bullet b|)\underset{w}{\prec }\lambda \left(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\right(A\left)\right)\ast \lambda \left(|b|\right)$, where A is a real symmetric positive semidefinite matrix and $A\bullet b$ is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality $\parallel a\circ b{\parallel }_p\le \parallel a{\parallel }_r\parallel b{\parallel }_s$, where $\parallel x{\parallel }_p:=\parallel \lambda \left(x\right){\parallel }_p$ denotes the spectral p-norm of x and $p,q,r\in [1,\mathrm{\infty }]$ with $\frac{1}{p}=\frac{1}{r}+\frac{1}{s}$. We also give precise values of the norms of the Lyapunov transformation $L_a$ and $P_a$ relative to two spectral p-norms.

受霍恩的对数多数（奇异值）不等式的启发$s\left(\mathrm{一个}乙\right)\underset{l○G}{\prec }s\left(\mathrm{一个}\right)*s\left(乙\right)$以及相关的弱多数化不等式$s\left(\mathrm{一个}乙\right)\underset{w}{\prec }s\left(\mathrm{一个}\right)*s\left(乙\right)$对于复方矩阵，我们考虑它们的 Hermitian 类似物$\lambda \left(\sqrt{\mathrm{一个}}乙\sqrt{\mathrm{一个}}\right)\underset{l○G}{\prec }\lambda \left(\mathrm{一个}\right)*\lambda \left(乙\right)$对于正半定矩阵和$\lambda (|\mathrm{一个}\circ 乙|)\underset{w}{\prec }\lambda \left(|\mathrm{一个}|\right)*\lambda \left(|乙|\right)$对于一般（Hermitian）矩阵，其中$\mathrm{一个}\circ 乙$表示A和B的 Jordan 积，$*$表示中的组件乘积${\mathcal{R}}^n$. 在本文中，我们将这些不等式扩展到欧几里德乔丹代数的设置，形式为$\lambda ({磷}_{\sqrt{\mathrm{一个}}}\left(b\right))\underset{l○G}{\prec }\lambda \left(\mathrm{一个}\right)*\lambda \left(b\right)$为了$\mathrm{一个},b\ge 0$和$\lambda (|\mathrm{一个}\circ b|)\underset{w}{\prec }\lambda \left(|\mathrm{一个}|\right)*\lambda \left(|b|\right)$对于所有a和b，其中${磷}_{你}$和$\lambda \left(你\right)$分别表示元素u的二次表示和特征值向量。我们还描述了形式的不等式$\lambda (|\mathrm{一个}\bullet b|)\underset{w}{\prec }\lambda \left(\mathrm{d}\mathrm{一世}\mathrm{一个}\mathrm{G}\right(\mathrm{一个}\left)\right)*\lambda \left(|b|\right)$，其中A是实对称半正定矩阵，并且$\mathrm{一个}\bullet b$是A和b的舒尔积。以应用的形式，我们证明了广义的 Hölder 型不等式$\parallel \mathrm{一个}\circ b{\parallel }_p\le \parallel \mathrm{一个}{\parallel }_r\parallel b{\parallel }_s$， 在哪里$\parallel X{\parallel }_p:=\parallel \lambda \left(X\right){\parallel }_p$表示x的谱p范数和$p,q,r\in [1,\mathrm{\infty }]$和$\frac{1}{p}=\frac{1}{r}+\frac{1}{s}$. 我们还给出了 Lyapunov 变换范数的精确值${\mathrm{大号}}_{\mathrm{一个}}$和${磷}_{\mathrm{一个}}$相对于两个光谱p范数。