Hardy Means of a Finite Blaschke Product and Its Derivative
Hardy Means of a Finite Blaschke Product and Its Derivative
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有限Blaschke积及其导数的Hardy均值
DOI：
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发表时间：
2013
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通讯作者：
F. Hartmann
中科院分区：
文献类型：
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作者：
Alan Gluchoff;F. Hartmann
在这一章中，我们考虑复平面单位圆盘上的有限Blaschke积(B_{n}(z)=prod_{k=1}^{n}frac{z_{k}-z}{1-overline{z}_{k}z})及其Hardy平均(M_{p}^{p}(r，B)=Frc{1}{2pi}int^{2pi}_{0}|B(Re^{i heta})|^{p}d heta)。我们讨论了(1-M_{2}^{2}(r，B))的两个显式公式：当B有不同的零点或单个零点重复n次时。我们将均值(M_{2}^{2}(r，B))和(M_{2}^{2}(r，B^{素数}))的增长与“抽样均值”((sum^{n}_{k=1}|B^{prime}(rz_{k})|(1-|z_{k}|^{2})).^{n}_{k=1}|B(rz_{k})|(1-|z_{k}|^{2}))和素数联系起来证明了对于二次和三次乘积，如果零点位于半径为|z|=ρ<；1的圆上，且连续零点之间的夹角ϕ不变，则(1-M_{2}^{2}(r，B))是ϕ的递增函数。我们推测，对于任意有限次的乘积，这一点也是成立的。
In this chapter we consider several topics related to finite Blaschke products (B_{n}(z)=prod_{k=1}^{n}frac{z_{k}-z}{1-overline{z}_{k} z}) in the unit disc of the complex plane and their Hardy means (M_{p}^{p}(r,B)=frac{1}{2pi}int^{2pi}_{0}|B(re^{i heta})|^{p} d heta). We discuss two explicit formulae for (1-M_{2}^{2}(r,B)): when B has distinct zeroes or a single zero repeated n times. We relate the growth of the means (M_{2}^{2}(r,B)) and (M_{2}^{2}(r,B^{prime})) to “sampling means” (sum^{n}_{k=1}|B(rz_{k})|(1-|z_{k}|^{2})) and (sum^{n}_{k=1}|B^{prime}(rz_{k})|(1-|z_{k}|^{2})). It is shown, for products of degree two and three, that if the zeroes lie on the circle of radius |z|=ρ<1 with constant angle ϕ between successive zeroes, then (1-M_{2}^{2}(r,B)) is an increasing function of ϕ. We conjecture that this holds true for products of arbitrary finite degree.