The article carried out a study on the convexity of the Bernatsky integral in the proposition that the function in question belongs to a subclass of star-shaped functions that satisfy certain conditions. For this, the condition of convexity of univalent functions was considered. The geometrical interpretation of the conditions is given, the radius of the convexity of the star-shaped functions is established. The intervals for the parameter are found for which the Bernatsky integral is a convex function in the whole unit circle, in cases where the parameter does not belong to the given interval, the Bernatsky integral will be a convex function in a circle of smaller radius. Three consequences are given in which various cases of convexity of the Bernatsky integral for analytic functions that belong to classes of functions with certain conditions are analyzed. For the considered classes of analytic functions, the radius of convexity of the Bernatsky integral is determined.