Szegő Type Asymptotics for Minimal Blaschke Products

Abstract

Let μ be a positive, finite Borel measure on [0,2π). For 0 <r< 1, 0 <p< ∞, let

$$En,p(d\mu ;r): = _{{B_n}}^{\inf }\left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {{{\left| {{B_n}\left( {r{e^{i\theta }}} \right)} \right|}^p}d\mu \left( \theta \right)} } \right\}1/p,$$

where the infimum is taken over all Blaschke products of ordernhaving zeros in |z| < 1. LetB ^* _n denote a minimal Blaschke product and letG(μ ^′) denote the geometric mean of the derivative of the absolutely continuous part ofμ. In the first part of the paper we present a self-contained proof of a result due to Parfenov; namelyE _n,p ~ r ⁿ G(μ′)^1/p as n → ∞. In the second part we describe the extension of the classical Szegő function D(z) and prove that B ^* _n (z) ~ z ⁿ G(µ′)^1/p/D(z)^2/p as n → ∞, uniformly on compact subsets of the annulus r < z < 1/r. Some generalizations and applications are also discussed.

Research was conducted while visiting the Institute or Constructive Mathematics, Department of Mathematics, University of south Florida.

Research supported in part ny the Nationl Science Foundation under grant DMS-881-4026.