Suppose $Z\in S_{++}^n$ (a symmetric positive definite $n\times n$ matrix), $V\in S^n$, $t\in\mathbb R$ such that $Z+tV\in\{X\in S_{++}^n\mid X\preceq 2Y\}$ where $Y\succeq 0$ is given. I need to show that the function:
$$ f(t)=-\log\det Z-\sum_{i=1}^n\log(1+t\lambda_i)-\text{trace}((Z+tV)^{-1}Y) $$
is concave in $t$. The $\lambda_i$ are eigenvalues of $Z^{-1/2}VZ^{-1/2}$. I know that I might be able to show this by showing that $f''(t)\le 0$ but the trace is stopping me from advancing - I don't know what to do with it. Any suggestions?
Hint for the trace term:
1) for fixed $Y$, the trace of $XY$ is linear, with derivative $Y$.
2) the derivative $A^{-1}$ with respect to $t$ is $-A^{-1}\tfrac{dA}{dt}A^{-1}$.
3) Combine 1) and 2) using the chain rule.