THEORY OF BERGMAN SPACES (Graduate Texts in Mathematics 199)

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135272/1/blms0246.pd

Estimation of Coefficients of Univalent Functions by a Tauberian Remainder Theorem

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135160/1/jlms0279.pd

Harmonic mappings onto stars

AbstractA general version of the Radó–Kneser–Choquet theorem implies that a piecewise constant sense-preserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings

On a theorem of Haimo regarding concave mappings

A relatively simple proof is given for Haimo’s theorem that a meromorphic function with suitably controlled Schwarzian derivative is a concave mapping. More easily verified conditions are found to imply Haimo’s criterion, which is now shown to be sharp. It is proved that Haimo’s functions map the unit disk onto the outside of an asymptotically conformal Jordan curve, thus ruling out the presence of corners

A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed for weighted Bergman spaces of analytic functions in the upper half-plane. The result is applied to show that the invariant subspaces of the shift operator on the standard Bergman space of the unit disk can be identified with those of a convolution Volterra operator on the space L2 (R+, (1/t)dt).Plan Nacional I+D (Ministerio de Ciencia y Tecnología)Junta de AndalucíaSecretaría de Estado de Educación y Universidade

Asymptotic zero distribution of hypergeometric polynomials

We show that the zeros of the hypergeometric polynomials , , cluster on the loop of the lemniscate {k mathord{left/{vphantom {k {left( {k + 1} right)}}} right.kern-nulldelimiterspace} {left( {k + 1} right)}}}}} right.kern-nulldelimiterspace} {left( {k + 1} right)^{k + 1} ,{text{Re}}left( z right) > {k mathord{left/{vphantom {k {left( {k + 1} right)}}} right.kern-nulldelimiterspace} {left( {k + 1} right)}}}}} right}$$]]> as infty$$]]> . We also state the equations of the curves on which the zeros of , lie asymptotically as infty$$]]> . Auxiliary results for the asymptotic zero distribution of other functions related to hypergeometric polynomials are proved, including Jacobi polynomials with varying parameters and associated Legendre functions. Graphical evidence is provided using Mathematica.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45435/1/11075_2004_Article_329322.pd