We are asked to show that the function $f(x,s,t) = -\log (st - \|x\|^2)$ is convex on the set $\{x \in \mathbb R^n, s > 0, t>0: \frac{\|x\|^2}{s}<t\}$
I tried proving this with derivatives but it very quickly gets out of hand, and the teacher said the solution is short and simple and not to look for verbose answers.
Another possible way is using composition of convex functions, but sadly that demands that the outer function is non decreasing, which $-\log$ is not.
Any ideas?