On the coefficients of 𝑝-valent functions which are polynomials of univalent functions

Todorov, Pavel G.

doi:10.1090/S0002-9939-1986-0845973-0

On the coefficients of $p$-valent functions which are polynomials of univalent functions
HTML articles powered by AMS MathViewer

by Pavel G. Todorov

Proc. Amer. Math. Soc. 97 (1986), 605-608

DOI: https://doi.org/10.1090/S0002-9939-1986-0845973-0

PDF | Request permission

Abstract:

We give explicit representations of the coefficients of $p$-valent functions which are polynomials of univalent functions of the class $S$. With their help we prove the Goodman conjecture in the special case that $f(z) = {[\varphi (z)]^p}$, $\varphi (z)$ in $S$. We also obtain sharp upper bounds for the coefficients of the considered $p$-valent functions in terms of the coefficients of the two component functions.

References

A. W. Goodman, On some determinants related to $p$-valent functions, Trans. Amer. Math. Soc. 63 (1948), 175–192. MR 23910, DOI 10.1090/S0002-9947-1948-0023910-X
Abdallah Lyzzaik and David Styer, Goodman’s conjecture and the coefficients of univalent functions, Proc. Amer. Math. Soc. 69 (1978), no. 1, 111–114. MR 460619, DOI 10.1090/S0002-9939-1978-0460619-7
Pavel G. Todorov, New explicit formulas for the coefficients of $p$-symmetric functions, Proc. Amer. Math. Soc. 77 (1979), no. 1, 81–86. MR 539635, DOI 10.1090/S0002-9939-1979-0539635-3
Pavel G. Todorov, Explicit formulas for the coefficients of Faber polynomials with respect to univalent functions of the class $\Sigma$, Proc. Amer. Math. Soc. 82 (1981), no. 3, 431–438. MR 612735, DOI 10.1090/S0002-9939-1981-0612735-4
Louis de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1-2, 137–152. MR 772434, DOI 10.1007/BF02392821