Records 
Author 
Galanopoulos, P.; Girela, D.; Hernandez, R. 
Title 
Univalent Functions, VMOA and Related Spaces 
Type 

Year 
2011 
Publication 
Journal Of Geometric Analysis 
Abbreviated Journal 
J. Geom. Anal. 
Volume 
21 
Issue 
3 
Pages 
665682 
Keywords 
Univalent functions; VMOA; Bloch function; Besov spaces; Logarithmic Bloch spaces; Logarithmic derivative; Schwarzian derivative; Smooth Jordan curve 
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). 
Address 
[Galanopoulos, P; Girela, D] Univ Malaga, Fac Ciencias, Dept Anal Matemat, E29071 Malaga, Spain, Email: galanopoulos_petros@yahoo.gr 
Corporate Author 

Thesis 

Publisher 
Springer 
Place of Publication 

Editor 

Language 
English 
Summary Language 

Original Title 

Series Editor 

Series Title 

Abbreviated Series Title 

Series Volume 

Series Issue 

Edition 

ISSN 
10506926 
ISBN 

Medium 

Area 

Expedition 

Conference 

Notes 
WOS:000291745600009 
Approved 

Call Number 
UAI @ eduardo.moreno @ 
Serial 
154 
Permanent link to this record 



Author 
Hernandez, R.; Martin, M.J. 
Title 
PreSchwarzian and Schwarzian Derivatives of Harmonic Mappings 
Type 

Year 
2015 
Publication 
Journal Of Geometric Analysis 
Abbreviated Journal 
J. Geom. Anal. 
Volume 
25 
Issue 
1 
Pages 
6491 
Keywords 
PreSchwarzian derivative; Schwarzian derivative; Harmonic mappings; Univalence; Becker's criterion; Convexity 
Abstract 
In this paper we introduce a definition of the preSchwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping f in the complex plane without assuming any additional condition on the (second complex) dilatation omega(f) of f. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove theorems analogous to those by Chuaqui, Duren, and Osgood. Also, we obtain a Beckertype criterion for the univalence of harmonic mappings. 
Address 
[Hernandez, Rodrigo] Univ Adolfo Ibanez, Fac Ingn & Ciencias, Vina Del Mar, Chile, Email: rodrigo.hernandez@uai.cl; 
Corporate Author 

Thesis 

Publisher 
Springer 
Place of Publication 

Editor 

Language 
English 
Summary Language 

Original Title 

Series Editor 

Series Title 

Abbreviated Series Title 

Series Volume 

Series Issue 

Edition 

ISSN 
10506926 
ISBN 

Medium 

Area 

Expedition 

Conference 

Notes 
WOS:000348344200003 
Approved 

Call Number 
UAI @ eduardo.moreno @ 
Serial 
452 
Permanent link to this record 