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Register a new userExtensions of Fiedler-Markhams inequality and Thompsons inequality
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We present some new inequalities related to determinant and trace for positive semidefinite block matrices by using symmetric tensor product, which are extensions of Fiedler-Markhams inequality and Thompsons inequality.
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In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality $$det(A^2+|BA|)le det(A^2+AB),$$ where $A, B$ are $ntimes n$ positive semidefinite matrices. We complement his result by proving $$det(A^2+|AB|)ge det(A^2+AB).$$ Our proofs feature the fruitful interplay between determinantal inequalities and majorization relations. Some related questions are mentioned.
We present a weak majorization inequality and apply it to prove eigenvalue and unitarily invariant norm extensions of a version of the Bohrs inequality due to Vasic and Kev{c}kic.
In this paper, we prove a Prekopa-Leindler type inequality of the $L_p$ Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Prekopa-Leindler inequality. In addition, we prove a functional $L_p$ Minkowski inequality.
An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d rightarrow infty$ is Lipschitz and $left{x_1, dots, x_N right} subset [0,1]^d$, then $$ left| int_{[0,1]^d} f(x) dx - frac{1}{N} sum_{k=1}^{N}{f(x_k)} right| leq left| abla f right|_{L^{infty}} cdot W_1left( frac{1}{N} sum_{k=1}^{N}{delta_{x_k}} , dxright),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Movers) Distance. We prove another such inequality with a smaller norm on $ abla f$ and a larger Wasserstein distance. Our inequality is sharp when the points are very regular, i.e. $W_{infty} sim N^{-1/d}$. This prompts the question whether these two inequalities are specific instances of an entire underlying family of estimates capturing a duality between transport distance and function space.
Yuan and Leng (2007) gave a generalization of Ky Fans determinantal inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski inequality $(det (A+B))^{1/n} ge (det A)^{1/n} +(det B)^{1/n}$, where $A$ and $B$ are positive semidefinite matrices. In this note, we first give an extension of Yuan-Lengs result to multiple positive definite matrices, and then we further extend the result to a larger class of matrices whose numerical ranges are contained in a sector. Our result improves a recent result of Liu [Linear Algebra Appl. 508 (2016) 206--213].
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