Univalent Functions, VMOA and Related Spaces

Abstract

This paper is concerned mainly with the logarithmic Bloch space ℬ_log which consists of those functions f which are analytic in the unit disc ${\mathbb{D}}$ and satisfy $\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty$, and the analytic Besov spaces B ^p, 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of:

• A bounded univalent function in $\bigcup_{p>1}B^{p}$ but not in the logarithmic Bloch space.

• A bounded univalent function in ℬ_log but not in any of the Besov spaces B ^p with p<2.

We also prove that the situation changes for certain subclasses of univalent functions. Namely, we prove that the convex univalent functions in ${\mathbb{D}}$ which belong to any of the spaces ℬ₀, VMOA, B ^p (1≤p<∞), ℬ_log, or some other related spaces are the same, the bounded ones.

We also consider the question of when the logarithm of the derivative, log g′, of a univalent function g belongs to Besov spaces. We prove that no condition on the growth of the Schwarzian derivative Sg of g guarantees log g′∈B ^p. On the other hand, we prove that the condition $\int _{{\mathbb{D}}}(1-\vert z\vert^{2})^{2p-2}\vert \mathit{Sg}(z)\vert ^{p}\,dA(z)<\infty$ implies that log g′∈B ^p and that this condition is sharp. We also study the question of finding geometric conditions on the image domain $g({\mathbb{D}})$ which imply that log g′ lies in B ^p. First, we observe that the condition of $g({\mathbb{D}})$ being a convex Jordan domain does not imply this. On the other hand, we extend results of Pommerenke and Warschawski, obtaining for every p∈(1,∞), a sharp condition on the smoothness of a Jordan curve Γ which implies that if g is a conformal mapping from ${\mathbb{D}}$ onto the inner domain of Γ, then log g′∈B ^p.