scholarly journals The Bloch Space and BMO Analytic Functions in the Tube over the Spherical Cone

1988 ◽  
Vol 102 (4) ◽  
pp. 949
Author(s):  
David Bekolle
Keyword(s):  
Analytic Functions ◽  
Bloch Space

On Analytic Functions of Bergman BMO in the Ball

1999 ◽  
Vol 42 (1) ◽  
pp. 97-103 ◽  
Author(s):  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Analytic Functions ◽  
Bloch Space ◽  
Bounded Mean Oscillation ◽  
Open Unit ◽  
Bergman Metric ◽  
Open Unit Ball ◽  
Mean Oscillation ◽  
Volume Measure

AbstractLet B = Bn be the open unit ball of Cn with volume measure v, U = B1 and B be the Bloch space on , 1 ≤ α < 1, is defined as the set of holomorphic f : B → C for whichif 0 < α < 1 and , the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic f : B → U for which the composition operator defined by , is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.

scholarly journals The Bloch space and Besov spaces of analytic functions

1996 ◽  
Vol 54 (2) ◽  
pp. 211-219 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Besov Spaces ◽  
Analytic Functions ◽  
Elementary Proof ◽  
Bloch Space ◽  
Little Bloch Space ◽  
Spaces Of Analytic Functions

We shall give an elementary proof of a characterisation for the Bloch space due to Holland and Walsh, and obtain analogous characterisations for the little Bloch space and Besov spaces of analytic functions on the unit disk in the complex plane.

scholarly journals Logarithmic Bloch space and its predual

2016 ◽  
Vol 100 (114) ◽  
pp. 1-16 ◽  
Author(s):  
Miroslav Pavlovic
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Bloch Space ◽  
Positive Coefficients ◽  
Little Bloch Space

We consider the space B1log?, of analytic functions on the unit disk D, defined by the requirement ?D|f?(z)|?(|z|) dA(z) < ?, where ?(r) = log?(1/(1?r)) and show that it is a predual of the ?log?-Bloch? space and the dual of the corresponding little Bloch space. We prove that a function f(z)=??n=0 an zn with an ? 0 is in B1 log? iff ??n=0 log?(n+2)/(n+1) < ? and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in B1 log?. Some properties of the Cesaro and the Libera operator are considered as well.

scholarly journals Characterizations of Bloch-Type Spaces of Harmonic Mappings

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Munirah Aljuaid ◽  
Flavia Colonna
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Bloch Space ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Space Of Analytic Functions ◽  
Type Spaces ◽  
Growth Space ◽  
Derivatives Of

We study the Banach space BHα (α>0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)<∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α>0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α>1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0<α<1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

Analytic functions with decreasing coefficients and Hardy and Bloch spaces

2012 ◽  
Vol 56 (2) ◽  
pp. 623-635 ◽  
Author(s):  
Miroslav Pavlović
Keyword(s):  
Analytic Functions ◽  
Bloch Space ◽  
Surprising Result ◽  
Bloch Spaces ◽  
Littlewood Theorem ◽  
Positive Coefficients

AbstractThe following rather surprising result is noted.(1) A function f(z) = ∑anzn such that an ↓ 0 (n → ∞) belongs to H1 if and only if ∑(an/(n + 1)) < ∞.A more subtle analysis is needed to prove that assertion (2) remains true if H1 is replaced by the predual, 1(⊂ H1), of the Bloch space. Assertion (1) extends the Hardy–Littlewood theorem, which says the following.(2) f belongs to Hp (1 < p < ∞) if and only if ∑(n + 1)p−2anp < ∞.A new proof of (2) is given and applications of (1) and (2) to the Libera transform of functions with positive coefficients are presented. The fact that the Libera operator does not map H1 to H1 is improved by proving that it does not map 1 into H1.

scholarly journals Multivalent functions andQKspaces

2004 ◽  
Vol 2004 (48) ◽  
pp. 2537-2546 ◽  
Author(s):  
Hasi Wulan
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Bloch Space ◽  
Univalent Functions ◽  
Nondecreasing Function ◽  
Invariant Space

We give a criterion forq-valent analytic functions in the unit disk to belong toQK, a Möbius-invariant space of functions analytic in the unit disk in the plane for a nondecreasing functionK:[0,∞)→[0,∞), and we show by an example that our condition is sharp. As corollaries, classical results on univalent functions, the Bloch space, BMOA, andQpspaces are obtained.

scholarly journals Hankel operators from the space of bounded analytic functions to the Bloch space

1999 ◽  
Vol 59 (1) ◽  
pp. 53-58 ◽  
Author(s):  
Ruhan Zhao
Keyword(s):  
Analytic Functions ◽  
Bloch Space ◽  
Hankel Operators ◽  
Bounded Analytic Functions ◽  
Little Bloch Space

Boundedness and compactness of little Hankel operators from H∞ to the Bloch space and the little Bloch space are characterised.

scholarly journals Bounded analytic functions and the little Bloch space

1990 ◽  
Vol 13 (1) ◽  
pp. 193-198 ◽  
Author(s):  
Rohan Attele
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Bloch Space ◽  
Bounded Analytic Functions ◽  
Radial Limits ◽  
Bounded Analytic Function ◽  
Little Bloch Space

The radial limits of the weighted derivative of an bounded analytic function is considered.

scholarly journals Besov-type characterisations for the Bloch space

1989 ◽  
Vol 39 (3) ◽  
pp. 405-420 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Ball ◽  
Besov Space ◽  
Analytic Functions ◽  
Unit Disc ◽  
Bloch Space ◽  
Bloch Functions ◽  
Little Bloch Space ◽  
Special Case

We will prove local and global Besov-type characterisations for the Bloch space and the little Bloch space. As a special case we obtain that the Bloch space consists of those analytic functions on the unit disc whose restrictions to pseudo-hyperbolic discs (of fixed pseudo-hyperbolic radius) uniformly belong to the Besov space. We also generalise the results to Bloch functions and little Bloch functions on the unit ball in . Finally we discuss the related spaces BMOA and VMOA.

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