Fractional iteration of functions analytic in the unit disk, with real coefficients

The present paper deals with the problem of fractional iteration in a class of analytic func- tions mapping the unit disk into itself, preserving the origin and having real coefficients of the expansion in a Maclaurin series, in terms of the Koenigs function. An integral representation of the class of Koenigs functions which correspond to the functions of the studied class admitting fractional iteration in this class is obtained. Some necessary conditions for the existence of frac- tional iterates of functions of the studied class in terms of estimates of their initial coefficients are given.