Abstract For a pair of conjugate trigonometrical polynomials C ( t ) = ∑ j = 1 N a j cos ⁡ j t , S ( t ) = ∑ j = 1 N a j sin ⁡ j t with real coefficients and normalization a 1 = 1 the following extremal value is found: sup a 2 , … , a N ⁡ min t ⁡ { C ( t ) : S ( t ) = 0 } = − 1 4 sec 2 ⁡ π N + 2 . An application of this result in geometric complex analysis is shown. Several conjectures for a number of extremal problems on classes of polynomials are suggested.

中文翻译：

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关于极值多项式族

摘要 对于一对共轭三角多项式 C ( t ) = ∑ j = 1 N aj cos ⁡ jt , S ( t ) = ∑ j = 1 N aj sin ⁡ jt 具有实系数和归一化 a 1 = 1 以下极值找到： sup a 2 , … , a N ⁡ min t ⁡ { C ( t ) : S ( t ) = 0 } = − 1 4 sec 2 ⁡ π N + 2 。显示了该结果在几何复数分析中的应用。建议了多项式类上许多极值问题的几个猜想。