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Motivated by Horn's log-majorization (singular value) inequality $s(AB)\underset{log}{\prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)\underset{w}{\prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $\lambda(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} \lambda(A)*\lambda(B)$ for positive semidefinite matrices and $\lambda(|A\circ B|) \underset{w}{\prec} \lambda(|A|)*\lambda(|B|)$ for general (Hermitian) matrices, where $A\circ B$ denotes the Jordan product of $A$ and $B$ and $*$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $\lambda\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} \lambda(a)*\lambda(b)$ for $a,b\geq 0$ and $\lambda\big (|a\circ b|\big )\underset{w}{\prec} \lambda(|a|)*\lambda(|b|)$ for all $a$ and $b$, where $P_u$ and $\lambda(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $\lambda(|A\bullet b|)\underset{w}{\prec} \lambda({\mathrm{diag}}(A))*\lambda(|b|)$, 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 H\"{o}lder type inequality $||a\circ b||_p\leq ||a||_r\,||b||_s$, where $||x||_p:=||\lambda(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,r\in [1,\infty]$ 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., Comment: 18 pages; abstract, introduction, and final section rewritten, references updated

A weighted complementarity problem (wCP) is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem (wHLCP) in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of R_0, R, and P properties and discuss the solvability of wHLCPs under nonzero (topological) degree conditions. A uniqueness result is stated in the setting of R^n. We show how our results naturally lead to interior point systems., Comment: 19 pages