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

Author:
Pavel G. Todorov

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

MSC:
Primary 30C50; Secondary 30C30

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

MathSciNet review:
845973

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Abstract: We give explicit representations of the coefficients of -valent functions which are polynomials of univalent functions of the class . With their help we prove the Goodman conjecture in the special case that , in . We also obtain sharp upper bounds for the coefficients of the considered -valent functions in terms of the coefficients of the two component functions.

**[1]**A. W. Goodman,*On some determinants related to**-valent functions*, Trans. Amer. Math. Soc.**63**(1948), 175-192. MR**0023910 (9:421g)****[2]**A. Lyzzaik and D. Styer,*Goodman's conjecture and the coefficients of univalent functions*, Proc. Amer. Math. Soc.**69**(1978), 111-114. MR**0460619 (57:612)****[3]**P. G. Todorov,*New explicit formulas for the coefficients of**-symmetric functions*, Proc. Amer. Math. Soc.**77**(1979), 81-86. MR**539635 (81g:30021)****[4]**-,*Explicit formulas for the coefficients of Faber polynomials with respect to univalent Junctions of the class*, Proc. Amer. Math. Soc.**82**(1981), 431-438. MR**612735 (82f:30016)****[5]**L. de Branges,*A proof of the Bieberbach conjecture*, Acta. Math.**154**(1985), 137-152. MR**772434 (86h:30026)**

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

Keywords:
-valent functions,
univalent functions,
the Goodman conjecture

Article copyright:
© Copyright 1986
American Mathematical Society