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An Extremal Problem for Odd Univalent Polynomials

Dmitrishin, Dmitriy; Gray, Daniel; Stokolos, Alexander; Tarasenko, Iryna

2023 Computational Methods and Function Theory

doi: 10.1007/s40315-023-00487-3

For the univalent polynomials F(z)=∑j=1Najz2j-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F(z) = \sum _{j=1}^{N} a_j z^{2j-1}$$\end{document} with real coefficients and normalization a1=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a_1 = 1$$\end{document} we solve the extremal problem minaj:a1=1-iF(i)=minaj:a1=1∑j=1N(-1)j+1aj.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \min _{a_j:\,a_1=1} \left( -iF(i) \right) = \min _{a_j:\,a_1=1} \sum _{j=1}^{N} {(-1)^{j+1} a_j}. \end{aligned}$$\end{document}We show that the solution is 12sec2π2N+2,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \frac{1}{2} \sec ^2\left( \frac{\pi }{2N+2}\right) , \end{aligned}$$\end{document}and the extremal polynomial ∑j=1NU2(N-j+1)′cosπ2N+2U2N′cosπ2N+2z2j-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \sum _{j = 1}^N \frac{U'_{2(N-j+1)} \left( \cos \left( \frac{\pi }{2N+2} \right) \right) }{U'_{2N} \left( \cos \left( \frac{\pi }{2N+2} \right) \right) }z^{2j-1} \end{aligned}$$\end{document}is unique and univalent, where Uj(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U_j(x)$$\end{document} is a Chebyshev polynomial of the second kind and Uj′(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U'_j(x)$$\end{document} denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in D\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {D}}$$\end{document} and formulate several conjectures.