Geodesic
Coefficient Regions for Univalent Trinomials
Canadian journal of mathematics, Tome 32 (1980) no. 1, pp. 1-20

Voir la notice de l'article provenant de la source Cambridge University Press

The problem of determining necessary and sufficient conditions bearing upon the numbers a 2 and a 3 in order that the polynomial z + a 2 z 2 + a 3 z 3 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. 
Rahman, Q. I.; Waniurski, J. Coefficient Regions for Univalent Trinomials. Canadian journal of mathematics, Tome 32 (1980) no. 1, pp. 1-20. doi: 10.4153/CJM-1980-001-0
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