scholarly journals A remark on boundedness of Bloch functions

1991 ◽  
Vol 44 (3) ◽  
pp. 527-528 ◽  
Author(s):  
Krzysztof Samotij
Keyword(s):  
Holomorphic Function ◽  
Unit Circle ◽  
Lebesgue Measure ◽  
Unit Disk ◽  
Complex Plane ◽  
Hausdorff Measure ◽  
Bloch Space ◽  
Bloch Functions ◽  
Zero Measure

Two consequences of a theorem of Dahlberg are derived. Let f be a holomorphic function in the unit disk D of the complex plane, and let E be an Fσ subset of the unit circle T. Suppose that |f(rw)| ≤ M, ω ∈ T/E, for some constant M.Then f is bounded in either of the two cases:(i) if f is in the Bloch space and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log log (2πee/t),(ii) if f is integrable with respect to the planar Lebesgue measure on D and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log(2πee/t).

scholarly journals On a Uniqueness Theorem

1969 ◽  
Vol 35 ◽  
pp. 151-157 ◽  
Author(s):  
V. I. Gavrilov
Keyword(s):  
Unit Circle ◽  
Uniqueness Theorem ◽  
Unit Disk ◽  
Complex Plane ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Extended Complex Plane

1. Let D be the open unit disk and r be the unit circle in the complex plane, and denote by Q the extended complex plane or the Rie-mann sphere.

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 Products of composition and n-th differentiation operators from α-bloch space to Qp space

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 761-766
Author(s):  
Haiying Li ◽  
Cui Wang ◽  
Tianyu Xue ◽  
Xiangbo Zhang
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Composition Operators ◽  
Bloch Space ◽  
Open Unit ◽  
Open Unit Disk ◽  
Th Differentiation ◽  
Qp Space

Let ? be an analytic self-map of the open unit disk D on the complex plane and ? > 0, p ? 0, n ? N. In this paper, the boundedness and compactness of the products of composition operators and nth differentiation operators C?Dn from a-Bloch space B? and B?0 to Qp space are investigated.

scholarly journals The three-separated-arc property of the modular function

1976 ◽  
Vol 61 ◽  
pp. 203-204
Author(s):  
Frederick Bagemihl
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Modular Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cluster Set

Let D be the open unit disk and Γ be the unit circle in the complex plane, and denote the Riemann sphere by Ω. If f(z) is a function defined on D with values belonging to Ω, if ζ ∈Γ, and if Λ is an arc at ζ then C∈(f, ζ) denotes the cluster set of f at ζ along Λ. If there exist three mutually exclusive arcs Λ1, Λ2, Λ3 at ζ such that

Qp Spaces on Riemann Surfaces

1998 ◽  
Vol 50 (3) ◽  
pp. 449-464 ◽  
Author(s):  
Rauno Aulaskari ◽  
Yuzan He ◽  
Juha Ristioja ◽  
Ruhan Zhao
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Bloch Space ◽  
Dirichlet Space ◽  
Function Spaces ◽  
Hyperbolic Riemann Surfaces ◽  
Qp Spaces

AbstractWe study the function spaces Qp(R) defined on a Riemann surface R, which were earlier introduced in the unit disk of the complex plane. The nesting property Qp(R) ⊆Qq(R) for 0 < p < q < ∞ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space AD(R) ⊆ Qp(R) for any p, 0 < p < ∞, thus sharpening T. Metzger's well-known result AD(R) ⊆ BMOA(R). Also the first author's result AD(R) ⊆ VMOA(R) for a regular Riemann surface R is sharpened by showing that, in fact, AD(R) ⊆ Qp,0(R) for all p, 0 < p < ∞. The relationships between Qp(R) and various generalizations of the Bloch space on R are considered. Finally we show that Qp(R) is a Banach space for 0 < p < ∞.

scholarly journals Extension of holomorphic L2-functions with weighted growth conditions

1992 ◽  
Vol 126 ◽  
pp. 141-157 ◽  
Author(s):  
Klas Diederich ◽  
Gregor Herbort
Keyword(s):  
Fixed Point ◽  
Holomorphic Function ◽  
Lebesgue Measure ◽  
Complex Plane ◽  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Small Neighborhood ◽  
Growth Conditions

In this article a new contribution to the following question is given: Let Ω ⊂ ⊂ Cn be a bounded pseudoconvex domain with C∞-smooth boundary, q ∈ ∂Ω a fixed point and H a k-dimensional affine complex plane such that q ∈ H and H intersects ∂Ω at q transversally. Let U be a suitably small neighborhood of q, and denote by r a C∞-defining function of Ω on U. Under which conditions on ∂Ω near q is it possible to find an exponent η>0 > 0 such that every holomorphic function f on Ω′ = H ∩Ω∩ U withwhere dλ′ denotes the Lebesgue-measure on H, can be extended to a holomorphic function ^f on Ω ∩ U such that even

scholarly journals A Property of the Principal Cluster sets of a Class of Holomorphic Functions

1971 ◽  
Vol 43 ◽  
pp. 157-159
Author(s):  
F. Bagemihl
Keyword(s):  
Meromorphic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cluster Set ◽  
Principal Cluster

Let D be the open unit disk and Γ be the unit circle in the complex plane, and denote by Ω the Riemann sphere. If f(z) is a meromorphic function in D, and if ζ∈Г, then the principal cluster set of f at ζ is the set

scholarly journals Intersections of Arc-Cluster Sets for Meromorphic Functions

1970 ◽  
Vol 40 ◽  
pp. 213-220 ◽  
Author(s):  
Charles L. Belna
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Limit Point ◽  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cluster Set ◽  
Jordan Arc

Let D and C denote the open unit disk and the unit circle in the complex plane, respectively; and let f be a function from D into the Riemann sphere Ω. An arc γ⊂D is said to be an arc at p∈C if γ∪{p} is a Jordan arc; and, for each t (0<t<1), the component of γ∩{z: t≤|z|<1} which has p as a limit point is said to be a terminal subarc of γ. If γ is an arc at p, the arc-cluster set C(f, p,γ) is the set of all points a∈Ω for which there exists a sequence {zk}a⊂γ with zk→p and f(zk)→a.

scholarly journals A Note on Extended Ambiguous Points

1971 ◽  
Vol 43 ◽  
pp. 167-168
Author(s):  
J.L. Stebbins
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Arbitrary Function ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk ◽  
Ambiguous Point

Let f be an arbitrary function from the open unit disk D of the complex plane into the Riemann sphere S. If p is any point on the unit circle C, C(f, p) is the set of all points w such that there exists in D a sequence of points {Zj} such that zj→p and f(zj)→w. CΔ(f, p) is defined in the same way, but the sequence {Zj} is restricted to Δ⊂D. If α and β are two arcs in D terminating at p and Cα(f, p)∩Cβ(f, p) = Φ, p is called an ambiguous point for f.

scholarly journals The three-arc and three-separated-arc properties of meromorphic functions

1974 ◽  
Vol 53 ◽  
pp. 137-140
Author(s):  
Frederick Bagemihl
Keyword(s):  
Meromorphic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Riemann Sphere ◽  
Open Unit ◽  
Open Unit Disk

Let Γ be the unit circle and D be the open unit disk in the complex plane, and denote the Riemann sphere by Ω. Suppose that f(z) is a meromorphic function in D, and that ζ ∈ Γ.

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