A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.

中文翻译：

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Blaschke积和超几何多项式的导数的极值

限于单位圆的有限Blaschke乘积是平滑的覆盖图。该图的导数的最大值和最小值反映了Blaschke乘积的几何形状。我们确定了两类极端的Blaschke乘积：那些使导数的最大值和最小值之间的差异最大的产品，以及使它们最小化的那些产品。两种类都具有相同的代数结构，即两个超几何多项式的商。