Roots of the equationf (z)=αf (α) for the class of typically-real functions

Abstract

Let Tr be the class of functionsf (z)=z+c2z2+..., regular in the disk ¦z¦ 0 in the remainder of the disk ¦z¦ <1. Let z f be the solution off (z)=αf (a) on Tr, whereα is any fixed complex numberα ≠ 0,α ≠ 1,α is any fixed real number, ¦α¦< 1. We determine the region\(D_{T_r } \) of values of the functional zf on the class Tr. Variation formulas for Stieltjes integrals due to G.M. Goluzin are used.