# Accession Number:

## AD0257057

# Title:

## 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

# Descriptive Note:

# Corporate Author:

## RICE UNIV HOUSTON TEX

# Personal Author(s):

# Report Date:

## 1961-04-01

# Pagination or Media Count:

## 1.0

# 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 Mr r these results are obtained via Caratheodorys theory of kernels. Author