Is $W(x)=(I-A\mathrm{diag}(\exp(x)) )^{-1}$ a log-convex function of $x$? Let $A$ be an $N \times N$ strictly substochastic matrix with all elements strictly positive, and let $W(x) = (I-A\mathrm{diag}(\exp(x)) )^{-1}$ with $x$ an $N$ dimensional vector with $x_i \leq 0,\forall i$. Is it the case that the entries of $W(x)$ are a log-convex function of $x$? It is easy to show that this is true if $A$ is diagonal.
 A: We want to show that $w_{ij}(x)$ is a log-convex
function of $x$. Note that
$$
\left(I-A\mathrm{diag}\left(\exp x\right)\right)^{-1}=I+A\mathrm{diag}\left(\exp x\right)+\left(A\mathrm{diag}\left(\exp x\right)\right)^{2}+...
$$
Invoking the result that the sum of log convex functions is
log convex, it is enough to show that the entries of
$$
B^{(n)}(x)=\left(A\mathrm{diag}\left(\exp x\right)\right)^{n}
$$
are log convex in $x$ when evaluated at any $x \leq 0$ and for $n=0,1,2,...$. Cases $n=0,1$ are trivial, while case $n=2$ entails
$$
b_{ij}^{(2)}(x)=\sum_{k}a_{ik}\exp x_{k}a_{kj}\exp x_{j}=\sum_{k}a_{ik}a_{kj}\exp x_{k}\exp x_{j}
$$
It is sufficient to show that each element in the sum is log convex,
hence it is enough to show that $\exp x_{k}\exp x_{j}$ is log convex,
which is trivial. Next, consider any other $n>2$ and imagine by an
induction argument that $B^{\left(n-1\right)}(x)$ is log convex.
We would have
$$
B^{(n)}(x)=B^{\left(n-1\right)}(x)A\mathrm{diag}\left(\exp x\right)
$$
and so
$$
b_{ij}^{\left(n\right)}(x)=\sum_{k}b_{ik}^{\left(n-1\right)}(x)a_{kj}\exp x_{j}
$$
It is again sufficient to show that
$$
b_{ik}^{\left(n-1\right)}(x)\exp x_{j}
$$
is log convex. This entails showing that
$$
x_{j}+\ln b_{ik}^{\left(n-1\right)}(x)
$$
is convex, which follows from the induction assumption.
