Power series with fast decreasing coefficients

Let $f(x) = \sum\limits_{n = 0}^\infty {{a_n}{x^n}}$ be an analytic function in the unit disk such that

$$ \left| {f(x)} \right| \leqslant {C_0}\exp \left( { - {C_1}{{\left| {\log \left( {1 - x} \right)} \right|}^\lambda }} \right),\quad \frac{1}{2} < x < 1, $$

and

$$ \left| {{a_n}} \right| \leqslant {C_2}\exp \left( { - {C_3}\frac{{\sqrt {n} }}{{\log \left( {n + 2} \right)}}} \right),\quad n \geqslant 0, $$

for some λ > 1, C ₀,C ₁,C ₂,C ₃ > 0. Then f ≡0. Bibliography: 5 titles.