An extremal problem for odd univalent polynomials

2022/08/03 by Dmitriy Dmitrishin, Daniel H.D. Gray, Dmitrishin, Dmitriy +5
Mathematics · #30C10 #30C25 #30C55 #30C75 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2208.02054

openalex publication_date 2022/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the univalent polynomials F(z) = ∑_j=1^N a_j 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} ∑_j=1^N (-1)^j+1 a_j. We show that the solution is \frac12 \sec²\fracπ2N+2, and the extremal polynomial ∑_{j = 1}^N \fracU'_2(N-j+1) ( cos(\fracπ2N+2))U'_2N ( cos(\fracπ2N+2))z^2j-1 is unique and univalent, where the U_j(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 \mathbb D and formulate several conjectures.

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