论文标题
奇数单价多项式的极端问题
An extremal problem for odd univalent polynomials
论文作者
论文摘要
对于单价多项式$ f（z）= \ sum \ limits_ {j = 1}^{n} a_j z^{2J-1} $带有真实系数和归一化\（a_1 = 1 \），我们解决了极端问题\ [ \ min_ {a_j：\，a_1 = 1} \ left（-if（i）\ right）= \ min_ {a_j：\，a_1 = 1} \ sum \ sum \ limits_ { \]我们证明该解决方案为$ \ frac12 \ sec^2 {\fracπ{2n+2}}}，$和极端多项式\ [ \ sum_ {j = 1}^n \ frac {u'_ {2（n-j+1）} \ left（\ cos \ left（\fracπ{2n+2} \ right）\ right）}} { \ cos \ left（\fracπ{2n+2} \ right）\ right）} z^{2J-1} \]是唯一的，无用的，其中$ u_j（x）$是第二种的chebyshev polynomials，而$ u'_j（x）$表示派生。作为应用程序，我们获得了$ \ Mathbb d $中奇数单价多项式的Koebe半径的估计，并提出了几种猜想。
For the univalent polynomials $F(z) = \sum\limits_{j=1}^{N} a_j z^{2j-1}$ 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 $\frac12 \sec^2{\fracπ{2N+2}},$ 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 $U'_j(x)$ 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.
