Univalent Functions, VMOA and Related Spaces

This paper is concerned mainly with the logarithmic Bloch space ℬlog  which consists of those functions f which are analytic in the unit disc \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{D}}$\end{document} and satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty $\end{document}, and the analytic Besov spaces Bp, 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of: A bounded univalent function in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\bigcup_{p>1}B^{p}$\end{document} but not in the logarithmic Bloch space.A bounded univalent function in ℬlog  but not in any of the Besov spaces Bp with p<2. We also prove that the situation changes for certain subclasses of univalent functions. Namely, we prove that the convex univalent functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{D}}$\end{document} which belong to any of the spaces ℬ0, VMOA, Bp (1≤p<∞), ℬlog , or some other related spaces are the same, the bounded ones.