and
given by power seriesUniversity of Kentucky
Abstract:
The Hadamard product of two analytic functions, and given by power series
and 
is
There is no clear geometric connection among the properties of 
and 
. The interesting paper of G.
Pólya and I.J. Schoenberg that appeared in the Pacific Journal of Mathematics in 1958 suggested that there are some
interesting connections. For example, they conjectured that the family of analytic functions that map the unit disk
onto convex domains is preserved under Hadamard products. This conjecture was proved by Ruscheweyh and
Sheil-Small in 1973. In this talk, we will survey the known geometric properties of Hadamard products. The talk
will include the definition and a discussion of "primes" under this product as well as some recent results of
Ruscheweyh and Suffridge. What does this have to do with space capsules? We will show that the if
maps the
unit disk onto an infinite strip and
maps the unit disk onto an infinite wedge, then
maps maps the disk
onto a region that resembles a space capsule.
Please join us for refreshments before the lecture at 2:30p.m. in room 353 Jabara Hall.
[ Fall 2001]