Univalent Functions in the Möbius Invariant QK Space

Abstract

It is shown that a univalent function f belongs to QK if and only if sup a∈𝔻∫01M∞2(r,f∘φa-f(a))K′(log (1/r))dr<∞, where φa(z)=(a-z)/(1-a¯z), provided K satisfies certain regularity conditions. It is also shown that under these conditions QK contains all univalent Bloch functions if and only if ∫01(log ((1+r)/(1-r)))2K′(log (1/r))dr<∞.