Аннотация:
For the univalent polynomials F(z) = P
N
j=1
aj z
2j−1 with real coefficients and
normalization a1 = 1 we solve the extremal problem
min
aj : a1=1
(−iF(i)) = min
aj : a1=1
X
N
j=1
(−1)j+1aj.
We show that the solution is 1
2
sec2 π
2N+2 , and the extremal polynomial
X
N
j=1
U
′
2(N−j+1)
cos
π
2N+2
U
′
2N
cos
π
2N+2 z
2j−1
is unique and univalent, where the Uj(x) are the Chebyshev polynomials of the second kind
and U
′
j
(x) denotes the derivative. As an application, we obtain the estimate of the Koebe
radius for the odd univalent polynomials in D and formulate several conjectures.
