Abstract
Let Tr be the class of functionsf (z)=z+c2z2+..., regular in the disk ¦z¦ <1, real on the diameter-1<z<1, and satisfying the condition Imf (z) · Im z>0 in the remainder of the disk ¦z¦ <1. Let z f be the solution off (z)=α f (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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
P. T. Mocanc, “On the equationf (z)=α (α) in the class of univalent functions,” Mathematica (Cluf),6, No. 1, 63–79 (1964).
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable [in Russian], Moscow (1966).
J. Kaczmarski, “Sur l'équationf (z)=pf (c) dans la classe des fonctions étoilees,” Eull. Acad. Polon. Sci.,15, No. 7, 455–464 (1967).
J. Kaczmarski, “Sur l'équationf (z)=pf (c) dans la famille des fonctions univalentes a coefficients réels,” Bull. Acad. Polon. Sci.,15, No. 4, 245–251 (1967).
L. Kh. Burshtein, “The roots of the equationf (z)=αf (α) for functions expressible by a Steiltjes integral,” Vestnik LGU, No. 13, 7–19 (1969).
N. A. Lebedev, “The majorant region for the expression in the class S},” Vestnik LGU,8, No. 3, 29–41 (1955).
I. A. Aleksandrov, “Extremal properties of the class S(w0),” Trudy Tomskogo Un-ta,169, 24–58 (1963).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 41–52, July, 1971.
Rights and permissions
About this article
Cite this article
Burshtein, L.K. Roots of the equationf (z)=αf (α) for the class of typically-real functions. Mathematical Notes of the Academy of Sciences of the USSR 10, 449–455 (1971). https://doi.org/10.1007/BF01747068
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01747068