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.