An Extremal Problem for Odd Univalent Polynomials

Abstract

For the univalent polynomials F(z)=∑j=1Najz2j-1 with real coefficients and normalization a1= 1 we solve the extremal problem minaj:a1=1(-iF(i))=minaj:a1=1∑j=1N(-1)j+1aj. We show that the solution is 12sec2(π2N+2), and the extremal polynomial ∑j=1NU2(N-j+1)′(cos(π2N+2))U2N′(cos(π2N+2))z2j-1 is unique and univalent, where Uj(x) is a Chebyshev polynomial of the second kind and Uj′(x) denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in D and formulate several conjectures. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.