Problem: another inequality

Let \(\mat{F},\mat{G}\) be positive definite matrices (do they have to be definite?) and \(0 \leq p \leq 2.\) Show that
\[
\tr\left(\mat{F}^{p/2} – \mat{G}^{p/2}\right) \leq \frac{p}{2} \tr\left((\mat{F} – \mat{G})\mat{G}^{p/2 – 1}\right).
\]

I’m working on it. See if you can get a proof before I do šŸ™‚

  • Pete

    Move terms around, differentiate with respect to F, (derivative is 0 when F=G). Use concavity of tr(F^(p/2)) to show it’s optimal. I think F can be semidefinite.

    • swiftset

      That seems to work. Good job Peter! You win a pumpkin.