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Crossposted from math.SE

Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $$\varphi (A) = -\text {tr}\ (A \log A).$$ Show that for all $t \in (0,1)$ $$\varphi ((1 - t) A + t B) \leq (1 - t) \varphi (A) + t \varphi (B) - \eta (t,1-t)$$ where $\eta (t,1 - t) = t \log (t) + (1 - t) \log (1 - t).$

I know that $\varphi$ is operator concave. But I don't have any idea as to how to bound $\varphi$ from above. Could anyone please give me some hint?

Thanks a bunch!

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  • $\begingroup$ This is a website for professional mathematicians to ask each other questions about their research. Homework problems are off-topic here. $\endgroup$
    – Nik Weaver
    Aug 30, 2021 at 4:22
  • $\begingroup$ @Nik Weaver this is not a homework problem. I have found it in the exercise of my book in matrix analysis. Could you please give me some hint? I have also asked it in MSE but couldn't get any response from anybody. So I thought that it could very well be related to research problem in matrix analysis. That's why I have posted it here. Thanks. $\endgroup$
    – RKC
    Aug 30, 2021 at 4:40

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