  Faybusovich and Tsuchiya [2017] draw a connection between matrix monotone
functions and self-concordant barriers. A convex $C^3$-smooth function $g$
whose derivative is matrix monotone on the positive semi-axis induces a convex
function $\varphi(W) = \mathop{\rm tr}(g(W))$ on the cone of squares of a
Euclidean Jordan algebra. For the epigraph set of $\varphi$, Faybusovich and
Tsuchiya [2017] derive a $(1 + d)$-self-concordant barrier, where $d$ is the
rank of the Jordan algebra. We propose a logarithmically homogeneous
self-concordant barrier with parameter $2 + d$ for the epigraph set of the
perspective of $\varphi$, which is a proper cone.
