Remark about the maximum of the modulus of an analytic function on the boundary

Let $\Lambda^{\alpha}$ be the analytic Hölder class in the unit disc $\mathbb D$. For $f\in \Lambda^{\alpha}$ and $I\subset\partial\mathbb D$, let $M_f(I)=\max_I|f|$. Assume that $I$, $J$ are arcs such that $|J|=2|I|$, $J$ and $I$ have common middle point. Then
$$ M_f(J)\le C(\alpha,f)\frac{|I|^{\alpha}+M_f(I)}{\log^{\alpha}\Bigl(\frac{|I|^{\alpha}}{M_f(I)}+2\Bigr)}. $$
It is proved that this estimate cannot be improved.