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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

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