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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 665-682
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)(1-|z|) log 1/1-|z| |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) (1-|z|(2))(2p-2) |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, E-29071 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 1050-6926 ISBN Medium
Area (up) Expedition Conference
Notes WOS:000291745600009 Approved
Call Number UAI @ eduardo.moreno @ Serial 154
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Author Efraimidis, I.; Gaona, J.; Hernandez, R.; Venegas, O.
Title On harmonic Bloch-type mappings Type
Year 2017 Publication Complex Variables And Elliptic Equations Abbreviated Journal Complex Var. Elliptic Equ.
Volume 62 Issue 8 Pages 1081-1092
Keywords Bloch functions; harmonic functions; Jacobian; univalent functions; schlicht radius; growth estimates; coefficient estimates; 30C25; 30C50; 30D45; 30H30
Abstract Let f be a complex-valued harmonicmapping defined in the unit disk D. We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which, roughly speaking, states that for. analytic log. is Bloch if and only if. is univalent.
Address [Efraimidis, I.] Univ Autonoma Madrid, Dept Matemat, Madrid, Spain, Email: iason.efraimidis@uam.es
Corporate Author Thesis
Publisher Taylor & Francis Ltd Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 1747-6933 ISBN Medium
Area (up) Expedition Conference
Notes WOS:000399938900004 Approved
Call Number UAI @ eduardo.moreno @ Serial 729
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Author Bravo, V.; Hernandez, R.; Venegas, O.
Title On the univalence of certain integral for harmonic mappings Type
Year 2017 Publication Journal Of Mathematical Analysis And Applications Abbreviated Journal J. Math. Anal. Appl.
Volume 455 Issue 1 Pages 381-388
Keywords Harmonic mapping; Univalent functions; Integral transform
Abstract We generalize the problem of univalence of the integral of f'(z)(alpha) when f is univalent to the complex harmonic mappings. To do this, we extend the univalence criterion by Ahlfors in [1] to those mappings. (C) 2017 Elsevier Inc. All rights reserved.
Address [Bravo, Victor] Univ Valparaiso, Inst Matemat, Fac Ciencias, Valparaiso, Chile, Email: victor.bravo@uv.cl;
Corporate Author Thesis
Publisher Academic Press Inc Elsevier Science Place of Publication Editor
Language English Summary Language Original Title
Series Editor Series Title Abbreviated Series Title
Series Volume Series Issue Edition
ISSN 0022-247x ISBN Medium
Area (up) Expedition Conference
Notes WOS:000424735900019 Approved
Call Number UAI @ eduardo.moreno @ Serial 826
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