Majorizations of Hadamard products of matrix powers (Q1375532)

For any positive definite \(n \times n\) matrix \(A\), the eigenvalues of \(A\) will be denoted \(\lambda _{1}(A) \geq \lambda _{2}(A) \geq \ldots \geq \lambda _{n}(A)\). The author proves a series of inequalities relating the eigenvalues of the Hadamard product \(A \circ B\) of two \(n \times n\) positive definite matrices \(A\) and \(B\) and the eigenvalues of various functions of \(A\) and \(B\). Specifically, for all real \(r\), \(0 < r < 1\), and \(k = 1,\ldots ,n\), he shows that \( \prod \lambda _{i}(A \circ B) \geq \prod \lambda _{i}(A^{r} \circ B^{r})^{1/r} \geq \prod e^{\lambda _{i} (I \circ (\log A + \log B))} \geq \prod e^{\lambda _{i} (\log A + \log B)} \geq \prod \lambda _{i}(A^{r}B^{r})^{1/r} \geq \prod \lambda _{i}(AB) \) \noindent where each product is over \(i = k,\ldots ,n\).