Method of variations for maps representable by means of Stieltjes integrals

In studying properties of classes of analytic functions and in finding extreme values of functionals on these classes, appropriate variational formulas are used. In this work, they are found for classes of functions that can be represented by a Stieltjes integral or some combination of such integrals. Variational formulas for classes of holomorphic functions with a positive real part in a circle are presented. Based on these formulas, variational formulas for star convex univalent mappings were obtained. Two variational formulas were obtained for functions holomorphic in a circular ring with a positive real part therein.