Schur-convex function

Abstract: Let $A$ and $B$ be $n\times n$ positive semidefinite Hermitian matrices, let $\alpha$ and $\beta$ be real numbers, let $\circ$ denote the Hadamard product of matrices, and let $A_{k}$ denote any $k\times k$ principal submatrix of $A.$ The following trace and eigenvalue inequalities are shown: $$ {\rm tr}(A\circ B)^\alpha \leq {\rm tr}(A^\alpha \circ B^\alpha ), \;\;\; \alpha \leq 0 \; \mbox{or} \; \alpha \geq 1,$$ $$ {\rm tr}(A\circ B)^\alpha \geq {\rm tr}(A^\alpha \circ B^\alpha ), \;\;\; 0\leq \alpha \leq 1,$$ $$ \lambda ^{1/\alpha }(A^\alpha \circ B^\alpha )\leq \lambda ^{1/\beta }(A^\beta \circ B^\beta ), \;\;\; \alpha \leq \beta , \alpha \beta eq 0,$$ $$ \lambda ^{1/\alpha }[(A^\alpha )_{k}]\leq \lambda ^{1/\beta }[(A^\beta )_{k}], \;\;\; \alpha \leq \beta , \alpha \beta eq 0.$$ The equalities corresponding to the inequalities above and the known inequalities $$ {\rm tr}(AB)^\alpha \leq {\rm tr}(A^\alpha B^\alpha ), \;\;\; |\alpha | \geq 1 ,$$ and $${\rm tr}(AB)^\alpha \geq {\rm tr}(A^\alpha B^\alpha ), \;\;\; |\alpha | \leq 1$$ are thoroughly discussed. Some applications are given.

Abstract: We characterize the (continuous) majorization of integrable functions introduced by Hardy, Littlewood, and Polya in terms of the (discrete) majorization of finite-dimensional vectors, introduced by the same authors. The most interesting version of this result is the characterization of the (increasing) convex order for integrable random variables in terms of majorization of vectors of expected order statistics. Such a result includes, as particular cases, previous results by Barlow and Proschan and by Alzaid and Proschan, and, in a sense, completes the picture of known results on order statistics. Applications to other stochastic orders are also briefly considered.

Abstract: Let $$m,n\in {\mathbb {N}}$$ and $$p\in (0,\infty )$$ . For a finite dimensional quasi-normed space $$X=({\mathbb {R}}^m, \Vert \cdot \Vert _X)$$ , let $$\begin{aligned} B_p^n(X) = \left\{ (x_1,\ldots ,x_n)\in \big ({\mathbb {R}}^{m}\big )^n: \sum _{i=1}^n \Vert x_i\Vert _X^p \leqslant 1\right\} . \end{aligned}$$ We show that for every $$p\in (0,2)$$ and X which admits an isometric embedding into $$L_p$$ , the function $$\begin{aligned} S^{n-1} i \uptheta = (\uptheta _1,\ldots ,\uptheta _n) \longmapsto \left| B_p^n(X) \cap \left\{ (x_1,\ldots ,x_n)\in \big ({\mathbb {R}}^{m}\big )^n: \sum _{i=1}^n \uptheta _i x_i=0 \right\} \right| \end{aligned}$$ is a Schur convex function of $$(\uptheta _1^2,\ldots ,\uptheta _n^2)$$ , where $$|\cdot |$$ denotes Lebesgue measure. In particular, it is minimized when $$\uptheta =\big (\frac{1}{\sqrt{n}},\ldots ,\frac{1}{\sqrt{n}}\big )$$ and maximized when $$\uptheta =(1,0,\ldots ,0)$$ . This is a consequence of a more general statement about Laplace transforms of norms of suitable Gaussian random vectors which also implies dual estimates for the mean width of projections of the polar body $$(B_p^n(X))^\circ $$ if the unit ball $$B_X$$ of X is in Lewis’ position. Finally, we prove a lower bound for the volume of projections of $$B_\infty ^n(X)$$ , where $$X=({\mathbb {R}}^m,\Vert \cdot \Vert _X)$$ is an arbitrary quasi-normed space.