Summability of the logarithm of an almost analytic function and a~generalization of the Levinson--Cartwright theorem

A. L. Vol'berg ; B. Jöricke

Sbornik. Mathematics, Tome 58 (1987) no. 2, pp. 337-349

Résumé

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.