
Galanopoulos, P., Girela, D., & Hernandez, R. (2011). Univalent Functions, VMOA and Related Spaces. J. Geom. Anal., 21(3), 665–682.
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 D and satisfy sup(z<1)(1z) log 1/1z f' (z)<infinity, and the analytic Besov spaces Bp, 1 <= p < infinity. 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 U(p>1) B(P) Bp 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 D which belong to any of the spaces B(0), VMOA, B(p) (1 <= p <= infinity), Blog, 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' is an element of B(p). On the other hand, we prove that the condition integral(D) (1z(2))(2p2) Sg(z)(p) d A(z)<infinity implies that log g' is an element of B(p) and that this condition is sharp. We also study the question of finding geometric conditions on the image domain g(D) which imply that log g' lies in Bp. First, we observe that the condition of g( 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 is an element of (1, infinity), a sharp condition on the smoothness of a Jordan curve Gamma which implies that if g is a conformal mapping from D onto the inner domain of Gamma, then log g' is an element of B(p).

