TY - JOUR

T1 - Univalent functions, VMOA and related spaces

AU - Galanopoulos, Petros

AU - Girela, Daniel

AU - Hernández, Rodrigo

N1 - Funding Information:
This research is partially supported by grants from “el Ministerio de Ciencia e Innovación, Spain” (MTM2007-60854, MTM2007-30904-E, MTM2008-0289-E and ‘Ingenio Mathematica (i-MATH)’ No. CSD2006-00032); from “La Junta de Andalucía” (FQM210, P06-FQM01504, and P09-FQM4468); from the European Networking Programme “HCAA” of the European Science Foundation; and by FONDECYT Grant ♯11070055, Chile.

PY - 2011/7

Y1 - 2011/7

N2 - 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(1<|z|) log 1/1|z| |f' (z)| <∞, 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 ∪ p>1 Bp 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 D which belong to any of the spaces B0, VMOA, Bp (1 ≤ p<∞), 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'ε Bp. On the other hand, we prove that the condition fD (1 - |z| 2)2p-2|Sg(z)|p dA(z)< ∞ implies that log g' ε Bp 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 ε (1,∞), a sharp condition on the smoothness of a Jordan curve γ which implies that if g is a conformal mapping from D onto the inner domain of γ, then log g' ε BpM.

AB - 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(1<|z|) log 1/1|z| |f' (z)| <∞, 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 ∪ p>1 Bp 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 D which belong to any of the spaces B0, VMOA, Bp (1 ≤ p<∞), 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'ε Bp. On the other hand, we prove that the condition fD (1 - |z| 2)2p-2|Sg(z)|p dA(z)< ∞ implies that log g' ε Bp 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 ε (1,∞), a sharp condition on the smoothness of a Jordan curve γ which implies that if g is a conformal mapping from D onto the inner domain of γ, then log g' ε BpM.

KW - Besov spaces

KW - Bloch function

KW - Logarithmic Bloch spaces

KW - Logarithmic derivative

KW - Schwarzian derivative

KW - Smooth Jordan curve

KW - Univalent functions

KW - VMOA

UR - http://www.scopus.com/inward/record.url?scp=80051799123&partnerID=8YFLogxK

U2 - 10.1007/s12220-010-9163-y

DO - 10.1007/s12220-010-9163-y

M3 - Article

AN - SCOPUS:80051799123

VL - 21

SP - 665

EP - 682

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 3

ER -