Prove $e^{e^{e^x}}$ is convex on $\mathbb{R}$.

Question

I know that we can prove $f(x)$ is convex by proving the inequality $e^{e^{e^{tx+(1-t)y}}}\le te^{e^{e^x}}+(1-t)e^{e^{e^y}} \forall x,y \in \mathbb{R}, t \in (0, 1)$, but I don't know where to start.

Answer 1

An increasing convex function of a convex function is convex. Just apply this repeatedly.