Geodesic
Modulus of boundary values of analytic functions of class~$\Lambda^n_\omega$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 258-260

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Let $\omega$ be a modulus of continuity, $\Lambda^n_\omega$ be the class of all functions analytic in the unit disk of the complex plane and such that $$ |f^{(n)}(z)-f^n(\zeta)|\le C_f\omega(|z-\zeta|)\quad(|z|,|\zeta|1). $$ A condition is given (depending essentially on $\omega$), necessary for a nonnegative function defined on the unit circle to coLncide with the modulus of some function of class $\Lambda^n_\omega$. 
@article{ZNSL_1981_113_a20,
     author = {N. A. Shirokov},
     title = {Modulus of boundary values of analytic functions of class~$\Lambda^n_\omega$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {258--260},
     publisher = {mathdoc},
     volume = {113},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_113_a20/}
}
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N. A. Shirokov. Modulus of boundary values of analytic functions of class~$\Lambda^n_\omega$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 258-260. http://geodesic.mathdoc.fr/item/ZNSL_1981_113_a20/