Geodesic
Functions with given estimate for $\partial f/\partial\overline z$, and N. Levinson's theorem
Sbornik. Mathematics, Tome 18 (1972) no. 2, pp. 181-189 
E. M. Dyn'kin. Functions with given estimate for $\partial f/\partial\overline z$, and N. Levinson's theorem. Sbornik. Mathematics, Tome 18 (1972) no. 2, pp. 181-189. http://geodesic.mathdoc.fr/item/SM_1972_18_2_a1/
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     title = {Functions with given estimate for $\partial f/\partial\overline z$, and {N.~Levinson's} theorem},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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. Bibliography: 7 titles.

[1] E. M. Dynkin, “Operatornoe ischislenie, osnovannoe na formule Koshi–Grina, i kvazianalitichnost klassov $D(h)$”, Zapiski nauchn. seminara LOMI, 19, 1970, 221–226 | MR | Zbl

[2] V. I. Matsaev, Nekotorye teoremy o polnote i kompaktnosti, svyazannye s klassicheskoi kvazianalitichnostyu, Dissertatsiya, Kharkov, 1964

[3] A. Zigmund, Trigonometricheskie ryady, Mir, Moskva, 1965 | MR

[4] S. Mandelbroit, Kvazianaliticheskie klassy funktsii, ONTI, Moskva, 1936

[5] N. Levinson, Gap and Density Theorems, Amer. Math. Soc. Coll. Publ., 26, American Mathematical Society, New York, 1940 | MR

[6] Y. Domar, “On a largest subharmonic minorant of a given function”, Arkiv Mat., 3:5 (1957), 429–440 | DOI | MR

[7] Zh. N. Keshishyan, “O nekotorykh kriteriyakh normalnosti semeistv analiticheskikh funktsii”, Izv. AN Arm. SSR, seriya fiz-matem., 15:2 (1962), 45–57 | MR