The mean Modulus of a Blaschke Product with Zeroes in a Nontangential Region
Titel:
The mean Modulus of a Blaschke Product with Zeroes in a Nontangential Region
Auteur:
Gluchoff, Alan
Verschenen in:
Complex variables and elliptic equations
Paginering:
Jaargang 1 (1983) nr. 4 pagina's 311-326
Jaar:
1983-06
Inhoud:
In this paper we investigate the behavior of a certain class of inner functions, i.e., functions which are bounded and analytic in the unit disc U and whose radial limits have modulus 1 almost everywhere. Recall that every inner function I can be written as[image omitted] where [image omitted] and σ is a finite positive Borel measure on [0,2π] singular with respect to Lebesgue measure. The first factor is called a Blaschke product, and the second is called a singular inner function; see [4] for details. We will deal with Blaschke products whose zeroes {zk} lie in a fixed nontangential region R(α) = {z ε U: (1 - |z|)/|1 - z|) > α} for some α > 0. Let C(α) be the class of all infinite Blaschke products with {zk} ⊂ R(α). We shall be concerned with the quantity[image omitted] Now △(r,φ ) → 0 as r → 1 for any inner function, and we shall determine bounds from above and below on △(r, B) for B ε C(alpha;), bounds which can be expressed explicitly in terms of the behavior of {zn}; this is done in part I. In part II we use the results of part I to get information on the derivatives of B ε C(α), their Taylor coefficients, and △(r, B) for {zn} with specific growth conditions. In what follows,[image omitted] Hp denotes the usual Hardy space of analytic functions on the unit diskLp[0, 1] the usual Lebesgue class on [0,1]. In addition, for 0 <p < 1Bp denotes the class of functions analytic in the unit disk for which[image omitted] Finally, the notation f(x) ≐ g(x) will be used to express the existence of constants C0, C1 > 0 such that C0g(x) ≤ f(x) ≤ C1, g(x) for all x in some domain which will be obvious from the context; C0 and C1 will be independent of x, though they may depend on other quantities. This paper is based on the author's thesis [6], supervised by Patrick Ahem at the University of Wisconsin, Madison.
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