Some norm inequalities involving convex functions of operators

Abstract

Let $A_{1},\ldots ,A_{n},X\in \mathbb {B(H)},$ where $A_{1},\ldots ,A_{n}$ are positive and $X\ge k I,$ for some positive real number k, let $\gamma _{1},\ldots ,\gamma _{n}$ be positive real numbers with $\sum \nolimits _{i=1}^{n}\gamma _{i}=1,$ and let $r\ge 2.$ We prove that

$$\begin{aligned} 2k {\left| \left| \left| \left( \sum \limits _{i=1}^{n}\gamma _{i}A_{i}\right) ^{r}+M^{r} \right| \right| \right| } \le {\left| \left| \left| \sum \limits _{i=1}^{n}\gamma _{i}(A_{i}^{r}X+XA_{i}^{r}) \right| \right| \right| }, \end{aligned}$$

for every unitarily invariant norm respecting submajorization, where $M^{2}=\sum \nolimits _{i\ne j}\frac{\gamma _{i}\gamma _{j}}{2}{\left| A_{i}-A_{j} \right| } ^{2}$ and I is the identity operator in $\mathbb { B(H)}$. We also give some related results.