Geodesic
Summability of the logarithm of an almost analytic function and a generalization of the Levinson–Cartwright theorem
Sbornik. Mathematics, Tome 58 (1987) no. 2, pp. 337-349 
A. L. Vol'berg; B. Jöricke. Summability of the logarithm of an almost analytic function and a generalization of the Levinson–Cartwright theorem. Sbornik. Mathematics, Tome 58 (1987) no. 2, pp. 337-349. http://geodesic.mathdoc.fr/item/SM_1987_58_2_a2/
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     title = {Summability of the logarithm of an almost analytic function and a~generalization of the {Levinson{\textendash}Cartwright} theorem},
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper is devoted to a generalization of a classical inequality: let $f$ be bounded and analytic in the disk $D$; then $f\not\equiv0\Rightarrow\int_{\mathrm{Fr}\mathbf D}\log|f(e^{i\theta})|\,d\theta>-\infty$, in the case of nonanalytic functions $f$. More precisely, it is proved that if $f=f_1+f_2$, where $f_1$ is the boundary function of a function of bounded characteristic, and $f_2$ is a function in a quasianalytic class (defined by some condition of regularity of decrease of its Fourier coefficients), then $\int_{\mathrm{Fr}\mathbf D}\log|f(e^{i\theta})|\,d\theta>-\infty$. The proof of this result depends in an essential way on a theorem of Levinson and Cartwright. At the same time, the result strengthens the Levinson–Cartwright theorem. Bibliography: 7 titles.

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[2] Beurling A., Quasianalyticity and general distributions, Stanford univ. lect. notes, 1961

[3] Volberg A. L., “Logarifm pochti analiticheskoi funktsii summiruem”, DAN SSSR, 265:6 (1982), 1297–1302 | MR

[4] Volberg A. L., Polnota ratsionalnykh drobei v vesovykh prostranstvakh na okruzhnosti, Dis. ... kand. fiz-mat. nauk, LOMI AN SSSR, L., 1983

[5] Beurling A., “Analytic continuation across a linear boundary”, Acta Math., 128:3–4 (1972), 153–182 | DOI | MR | Zbl

[6] Dynkin E. M., “Funktsii s dannoi otsenkoi $\partial f/\partial\overline z$ i teorema N. Levinsona”, Matem. sb., 89(131):2(10) (1972), 182–190 | MR | Zbl

[7] Brelo M., Osnovy klassicheskoi teorii potentsiala, Mir, M., 1964 | MR | Zbl