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- W206013558 abstract "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^{itheta})|^{p} dtheta). 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." @default.
- W206013558 created "2016-06-24" @default.
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- W206013558 date "2013-01-01" @default.
- W206013558 modified "2023-09-26" @default.
- W206013558 title "Hardy Means of a Finite Blaschke Product and Its Derivative" @default.
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- W206013558 doi "https://doi.org/10.1007/978-1-4614-5341-3_9" @default.
- W206013558 hasPublicationYear "2013" @default.
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