Power series with fast decreasing coefficients

Let $f(x)=\sum_{n=0}^\infty a_nx^n$ be an analytic function in the unit disc such that for some $\lambda>1$, $C_0,C_1,C_2,C_3>0$ we have
$$ |f(x)|\le C_0\exp(-C_1|\log(1-x)|^\lambda),\qquad\frac12<x<1 $$
and
$$|a_n|\le C_2\exp\biggl(-C_3\frac{\sqrt n}{\log(n+2)}\biggr),\qquad n\ge0. $$
Then $f\equiv0$. Bibl. – 5 titles.