Functions with given estimate for $\partial f/\partial\overline z$, and N. Levinson's theorem

In this paper it is shown that a twice continuously differentiable function $\varphi$ on the unit circle with Fourier coefficients $\{\widehat\varphi(n)\}$ admits a continuously differentiable extension $f$ to the whole plane such that
$$ \frac{\partial f}{\partial\overline z}=O[h(|1-|z||)] $$
(here $h$ is a given weight with $h(+0)=0)$ if $\varphi(n)=O(n^{-1}a_n)$, where
$$ a_n=\int_0^1h(r)(1-r)^{|n|}\,dr,\qquad n=0,\pm1,\pm2,\dots\,. $$

If $\int_0\ln\ln\frac1{h(r)}\,dr<+\infty$, then the class of such functions $\varphi$ turns out to be non-quasi-analytic. Hence a new proof of the known theorem of N. Levinson on the normality of families of analytic functions is derived.
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