About sharpness of the estimate in a theorem concerning half smoothness of a function holomorphic in a ball

Let $\mathbb B^n$ be the unit ball and $S^n$ the unit sphere in $\mathbb C^n$, $n\geq2$. Take $\alpha$, $0<\alpha<1$, and define a function $f$ on $\overline{\mathbb B^n}$ as follows:
$$ f(z)= (z_1-1)^\alpha e^{\frac{z_1+1}{z_1-1}},\quad z=(z_1,\dots,z_n)\in\overline{\mathbb B^n}. $$
The main result of the paper is the following.
Theorem. {\it If considered on the unit sphere $S^n$, the function $\zeta\mapsto|f(\zeta)|$ belongs to the Hölder class $H^\alpha(S^n)$; the function $f$ does not belong to the Hölder class $H^{\frac\alpha2+\varepsilon}(\overline{\mathbb B^n})$ for any $\varepsilon>0$.}