Let $f_a:\mathbb{N}\rightarrow \mathbb{R}$, with $f_a(n)=(\log n/\log\log n)^a,$ and $a>0.$ Let $g_a(n)=f_a(n) 2^{f_a(n)}.$ It seems that for increasing $a,$ somewhere around $a=a^{\ast},$ with $a^{\ast}=1.672\cdots$, $g_a(n)$ becomes convex-$\cup$ while it is convex-$\cap$ for $a\in (0,a^{\ast}).$
What's special about this value $a^{\ast},$ is it related to a known/important constant?