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Coefficient Regions for Univalent Trinomials

Published online by Cambridge University Press:  20 November 2018

Q. I. Rahman  and
J. Waniurski
Q. I. Rahman
Affiliation:
Université de Montréal, Montréal, Québec
J. Waniurski
Affiliation:
Maria Curie-Sklodowska University, Lublin, Poland
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The problem of determining necessary and sufficient conditions bearing upon the numbers a2 and a3 in order that the polynomial z + a2z2 + a3z3 be univalent in the unit disk |z| < 1 was solved by Brannan ([3], [4]) and by Cowling and Royster [6], at about the same time. For his investigation Brannan used the following result due to Dieudonné [7] and the well-known Cohn rule [9].

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, (2) is to be interpreted as the derivative of (1).

The procedure of Cowling and Royster was based on the observation that is univalent in |z| < 1 if and only if for all α such that 0 ≧ |α| ≧ 1, α ≠ 1 the function

is regular in the unit disk.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 1 , 01 February 1980 , pp. 1 - 20
Copyright
Copyright © Canadian Mathematical Society 1980

References

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