Concavity of matrix logarithm I am looking for the proof of the following concave like property of the logarithm operator. 
Let   $\bf{0} \preceq {\bf A} \preceq {\bf I}$. Then, for any positive definite matrices $ {\bf \Sigma}_1, {\bf \Sigma}_2$
\begin{align}
\log \left(   ({\bf I} - {\bf A} )^{\frac{1}{2}}  {\bf \Sigma}_1 ({\bf I} - {\bf A} )^{\frac{1}{2}}   +  {\bf A} ^{\frac{1}{2}}  {\bf \Sigma}_2  {\bf A}^{\frac{1}{2}} \right) \succeq    ({\bf I} - {\bf A} )^{\frac{1}{2}} \log ( {\bf \Sigma}_1 ) ({\bf I} - {\bf A} )^{\frac{1}{2}} +   {\bf A} ^{\frac{1}{2}} \log ( {\bf \Sigma}_2) {\bf A} ^{\frac{1}{2}}.
\end{align}
Note that this is true for $n=1$ since
\begin{align}
\log( (1-\alpha) \sigma_1+\alpha \sigma_2) \ge  (1-\alpha) \log(\sigma_1)+\alpha \log(\sigma_2)
\end{align}
A reference would be greatly appreciated. 
 A: The stated inequality is basically a special case of Hansen and Pedersen's characterisation of operator convex functions. See section 2.5 of

Fumio Hiai, Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization, Interdisciplinary Information Sciences, vol. 16, no. 2 (2010) 139-248 (https://doi.org/10.4036/iis.2010.139)

or

Pattrawut Chansangiam, A Survey on Operator Monotonicity, Operator Convexity, and Operator Means, International Journal of Analysis, vol. 2015 (http://dx.doi.org/10.1155/2015/649839)

It is well known that $\log:(0,\infty)\to\mathbb R$ is operator concave (this is also proved in example 13(iii) of Chansangiam's survey paper). It follows that for every $\epsilon>0$, the function $f_\epsilon(t):[\epsilon,\infty)\to\mathbb R$ defined by $t\mapsto-\log(t+\epsilon)$ is operator convex. Now, by Hansen-Pedersen characterisation (equivalence of conditions (i) and (v) in theorem 2.5.2 of Hiai or theorem 9 or Chansangiam), the stated inequality in your question holds when $\log$ is replaced by $-f_\epsilon$. Let $\epsilon\to0$, the conclusion follows.
