An Extremal Problem for Odd Univalent Polynomials

Abstract

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.