Reverse Holder, Minkowski, And Hanner Inequalities For Matrices

Abstract

We examine a number of known inequalities for $L^p$ functions with reverse representations for $s<1$ with complex matrices under the $p$-norms $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$, and similarly defined quasinorm or antinorm quantities $||X||_s=\text{Tr}[(X^\ast X)^{s/2}]^{1/s}$. Analogous to the reverse H\"{o}lder and reverse Minkowski for $L^p$ functions, it has recently been shown that for $A,B\in M_{n\times n}(\mathbb{C})$ such that $|B|$ is invertible, $||AB||_1\geq ||A||_{s}||B||_{s/(s-1)}$ and for $A,B$ positive semidefinite that $||A+B||_s\geq ||A||_s+||B||_s$. We comment on variational representations of these inequalities. A third very important inequality is Hanner's inequality $||f+g||_p^p+||f-g||_p^p\geq(||f||_p+||g||_p)^p+|||f||_p-||g||_p|^p$ in the $1\leq p\leq 2$ range, with the inequality reversing for $p\geq 2$. The analogue inequality has been proven to hold matrices in certain special cases. No reverse Hanner has established for functions or matrices considering ranges with $s<1$. We develop a reverse Hanner inequality for functions, and show that it holds for matrices under special conditions; it is sufficient but not necessary for $C+D, C-D\geq 0$. We also extend certain related singular value rearrangement inequalities that were previously known in the $1\leq p\leq3$ range to the $s<1$ range. Finally, we use the same techniques to characterize the previously unstudied equality case: we show that there is equality when $p\neq 1,2$ if and only if $|D|=k|C|$, which is directly analogous to the $L^p$ equality condition.