Löwner expansions

Abstract

A fundamental problem is to estimate the logarithmic coefficients of a power series with constant coefficient zero which represents a function which has distinct values at distinct points of the unit disk. A source of estimates is an expansion theorem for the Löwner equations which is obtained from a study of contractive substitutions in Hilbert spaces of analytic functions. The methods are an outgrowth of the theory of square summable power series [1]. Assume that σ n is a given function of nonnegative integers n, with nonnegative values, such that σ 0 = 0 and such that σ n − 1 ⩽ σ n when n is positive. Infinite values are allowed. The underlying Hilbert space is the set G σ(0) of equivalence classes of power series f( z) = ∑ a n z n with constant coefficient zero such that  f( z) 2 G σ (0) = ∑( n/σ n)| a n| 2 is finite. Equivalence of power series f( z) and g( z) means that the coefficient of z n in f( z) is equal to the coefficient of z n in g( z) when σ n is finite.