On the zeros of analytic functions belonging to Gevrey classes

For functions $f(z)\not\equiv0$, holomorphic in the unit disk $u$, infinitely differentiable in $\overline u$, and belonging to a given $\partial u$ class on partu, we establish sufficient conditions characterizing the sets
$$ K_f^\infty=\{z:|z|=1,f^{(k)}(z)=0,\quad k=0,1,2,\dots\}. $$
These conditions are close to the necessary condition due to L. Carleson and substantially more precise than the conditions given by A.-M. Chollet (see [1, 2]).