A difference counterpart to a matrix Hölder inequality
Let \(A,A_1,\dots,A_n,B,B_1,\dots B_n\) be Hermitian positive definite matrices of the same dimension. Assume that there are real numbers \(0<m\leq M\) such that (under the Löwner partial ordering) \(0<mA_i\leq B_i\leq MA_i\) for all~\(i=1,\dots,n\). Define the geometric mean of \(A\) and \(B\) by~\(A\,\sharp\,B= A^\frac{1}{2} (A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^\frac{1}{2}A^\frac{1}{2}\). The authors prove that the matrix Cauchy-Schwarz inequality has a ``difference type reverse'' \[ \big(\sum_{i=1}^nA_i\big)\,\sharp\,\big(\sum_{i=1}^nB_i\big)- \sum_{i=1}^nA_i\,\sharp\,B_i\leq \frac{(\sqrt{M}-\sqrt{m})^2}{4(\sqrt{M}+\sqrt{m})} \sum_{i=1}^nA_i \] and, more generally, that the matrix Hölder inequality has such a reverse \[ \big(\sum_{i=1}^nA_i\big)\,\sharp_\alpha\,\big(\sum_{i=1}^nB_i\big)- \sum_{i=1}^nA_i\,\sharp_\alpha\,B_i\leq -C(m,M,\alpha) \sum_{i=1}^nA_i. \] Here \(0\leq\alpha\leq 1\), \(A\,\sharp_\alpha\,B= A^\frac{1}{2}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^\alpha A^\frac{1}{2}\), and \[ C(m,M,\alpha)=(\alpha-1)\Big(\frac{M^\alpha-m^\alpha} {\alpha(M-m)}\Big)^\frac{\alpha}{\alpha-1}+ \frac{Mm^\alpha-mM^\alpha}{M-m}. \] (The first summand should be regarded as the corresponding limit if \(\alpha=0\) or \(\alpha=1\).) These results follow from more general lemmas on positive maps between matrix spaces. As an application, the authors prove that the relative operator entropy \[ S(A\,|\,B)=A^\frac{1}{2} (\log A^{-\frac{1}{2}}BA^{-\frac{1}{2}})A^\frac{1}{2}= \lim_{\alpha\to 0}\frac{A\,\sharp_\alpha\,B-A}{\alpha}, \] due to \textit{J.~I.~Fujii} and \textit{E.~Kamei} [Math. Japon. 34, 341--348 (1989; Zbl 0699.46048)], satisfies \[ S\big(\sum_{i=1}^nA_i\,|\,\sum_{i=1}^nB_i\big)- \sum_{i=1}^nS(A_i\,|\,B_i)\leq\log{S(h)\sum_{i=1}^nA_i}. \] Here the Specht ratio \[ S(h)=\frac{(h-1)h^\frac{1}{h-1}}{\mathrm{e}\log{h}}\,\, (h\neq 1),\quad S(1)=1,\quad h=\frac{M}{m}. \]
scientific article; zbMATH DE number 5689422