How to prove function $f(x,y)=y^xe^{-y}$ in $x \in R_+^n$ for all $ y \gt 0$ is both logarithmically concave and logarithmically convex(i.e. $\log f(x,y)=x \log y-y$ is both a concave function and a convex function)?
My problem comes from solutions manual to Convex Optimizaton, exercise 3.52 at page 69
It says “log-convex (as well as log-concave) in $x$”, which means that if we keep $y$ constant, then $\log f(x,y)$ is an affine function of $x$, which is in fact the case.