An extremal problem for odd univalent polynomials

Abstract

For the univalent polynomials $F(z)=\sum _{j=1}^N{a}_j{z}^{2j-1}$ with real coefficients and normalization \(a_1 = 1\) we solve the extremal problem \[ \min_{a_j:\,a_1=1} \left( -iF(i) \right) = \min_{a_j:\,a_1=1} \sum\limits_{j=1}^{N} {(-1)^{j+1} a_j}. \] We show that the solution is $\frac{1}{2}{\sec }^2\frac{\pi }{2N+2},$ and the extremal polynomial \[ \sum_{j = 1}^N \frac{U'_{2(N-j+1)} \left( \cos\left(\fracπ{2N+2}\right)\right)}{U'_{2N} \left( \cos\left(\fracπ{2N+2}\right)\right)}z^{2j-1} \] is unique and univalent, where the $U_j(x)$ are the Chebyshev polynomials of the second kind and $U_j^{\prime }(x)$ 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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