Matrix inequalities related to Hölder inequality

Abstract

Matrix inequalities of Hölder type are obtained. Among other inequalities, it is shown that if $p,q \in (2,\infty) $ and $r>1$ with $1/p+1/q=1-1/r$, then for any $A_{i},B_{i}\in M_{n}\left(\mathbb{C} \right) $ and $\alpha _{i}\in \left[ 0,1\right] $ $\left( i=1,2,\cdots ,m\right) $ with $\sum\limits_{i=1}^{m}\alpha _{i}=1$, we have% \begin{equation*} \left\vert \sum\limits_{i=1}^{m}\alpha _{i}^{1/r}B_{i}A_{i}\right\vert \leq \left( \sum\limits_{i=1}^{m}\left\vert A_{i}\right\vert ^{p}\right) ^{1/p} \end{equation*}% whenever $\sum\limits_{i=1}^{m}\left\vert B_{i}^{\ast }\right\vert ^{q}\leq I $. Related unitarily invariant norm inequalities are also presented.