ON THE GEOMETRY OF FUNCTIONS HOLOMORPHIC IN THE UNIT CIRCLE, OF ARBITRARILY SLOW GROWTH, WHICH TEND TO INFINITY ON A SEQUENCE OF CURVES APPROACHING THE CIRCUMFERENCE
Abstract
IT IS WELL KNOWN THAT THERE EXIST FUNCTIONS H( ), holomorphic in 1, with H( ) ( ) where (r) is a given positive function which as r 1, and such that min rn H( ) approaches as n . Here rn 1 is an appropriately chosen sequence. Such functions may be constructed by the use of gap series or via an infinite product. The object of the present note is to construct such a function geometrically by starting with the Riemann surface onto which w = H( )maps 1. The essence of the argument is in showing that is hyperbolic and that M(r) (r); these results are obtained via Caratheodory's theory of kernels. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1961
- Accession Number
- AD0257057
Entities
People
- Gerald R. Maclane
Organizations
- Rice University