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On Some Classes of Univalent Polynomials

Published online by Cambridge University Press:  20 November 2018

Q. I. Rahman  and
J. Szynal
Q. I. Rahman
Affiliation:
University of Montreal, Montreal, Quebec
J. Szynal
Affiliation:
University of Montreal, Montreal, Quebec
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It was in the year 1931 that Dieudonné [4] proved the following necessary and sufficient condition for a polynomial to be univalent in the unit disk.

THEOREM A (Dieudonné criterion). The polynomial

(1)

is univalent in |z| < 1 if and only if for every θ in [0, π/2] the associated polynomial

(2)

does not vanish in |z| < 1. For θ = 0, ϕ(z, θ) is to be interpreted as Pn'(z).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 2 , 01 April 1978 , pp. 332 - 349
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Brannan, D. A., On univalent polynomials and related classes of functions, Thesis, University of London, 1967.Google Scholar
2. Brannan, D. A., Coefficient regions for univalent polynomials of small degree, Mathematika 14 (1967), 165–169.Google Scholar
3. Cowling, V. F. and Royster, W. C., Domains of variability for univalent polynomials, Proc. Amer. Math. Soc. 19 (1968), 767–772.Google Scholar
4. Dieudonné, J., Recherches sur quelques problèmes relatifs aux polynômes et aux fonctions bornées d'une variable complexe, Ann. Ecole Norm. Sup. (3) 1+8 (1931), 247–358.Google Scholar
5. Hayman, W. K., Multivalent functions (Cambridge University Press, 1958).Google Scholar
6. Krzyz, J. and Rahman, Q. I., Univalent polynomials of small degree, Ann. Univ. M. Curie- Skłodowska, Sec. A 21 (1967), 79–90.Google Scholar
7. Marden, M., Geometry of polynomials, Amer. Math. Soc. Math. Surveys, 3 (1966).Google Scholar
8. Michel, C., Eine Bemerkung zu schlichten Polynomen, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 18 (1970), 513–519.Google Scholar
9. Ruscheweyh, St. and Wirths, K. J., Ûber die Koeffizienten spezieller schlichter Polynôme, Ann. Polon. Math. 28 (1973), 341–355.Google Scholar
10. Suffridge, T. J., On univalent polynomials, J. London Math. Soc. 44 (1969), 496–504.Google Scholar
11. Suffridge, T. J., Extreme points in a class of polynomials having univalent sequential limits, Trans. Amer. Math. Soc. 163 (1972), 225–237.Google Scholar