Show that the function is convex

Question

Show that the function $f: S \to \mathbb R$ given by $$f(x,s,t):=-\ln(st – ||x||^2)$$ is convex on $$S := \left\{(x,s,t) \in \mathbb R^n \times \mathbb R \times \mathbb R: \frac{\|x\|^2}{s}<t, s>0, t>0 \right\}$$

$-\ln(x)$ is a convex monotonic decreasing function, hence we can use composition by showing the inner part is concave. Also we know $-||x||^2$ is concave and $st$ is neither convex nor concave on $S$. Also in $S$ we have $st>||x||^2$ but I was unable to show the convexity of the function as a whole

Answer 1

I think you can try the following:

$-\ln(st-\| x\|^2)=-\ln(s(t-\frac{| x\|}{s}^2))=-\ln(s)-\ln(t-\frac{| x\|}{s}^2)$

Then you have a sum of two convex functions.