Modulus of boundary values of analytic functions of class $\Lambda^n_\omega$

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 $\Lambda^n_\omega$.