  The aim of this article is to analyze some peculiar features of the global
(and local) minima of $\alpha$-Brjuno functions $B_\alpha$ where
$\alpha\in[\frac{1}{2},1].$ Our starting point is the result by
Balazard--Martin (2020), who showed that the minimum of $B_1$ is attained at
$g:=\frac{\sqrt 5 -1}{2}$; analyzing the scaling properties of $B_1$ near $g$
we shall deduce that all preimages of $g$ under the Gauss map are also local
minima for $B_1$. Next we consider the problem of characterizing global and
local minima of $B_\alpha$ for other values of $\alpha$: we show that for
$\alpha\in (g,1)$ the global minimum is again attained at $g$, while for
$\alpha=1/2$ the function $B_{1/2}$ attains its minimum at
$\gamma:=\sqrt{2}-1$.
