Holomorphic maps of discs into balls of lp-spaces

1986 ◽  
Vol 99 (1) ◽  
pp. 123-133
Author(s):  
Josip Globevnik
Keyword(s):  
Unit Ball ◽  
Unit Disc ◽  
Open Unit ◽  
Holomorphic Maps ◽  
Open Unit Disc ◽  
Open Unit Ball ◽  
Holomorphic Map ◽  
Lp Spaces

AbstractLet 1 ≤ p < ∞, let be the open unit ball of lp and let Δ be the open unit disc in ℂ. We prove that there is a holomorphic map such that F(Δ) is dense in and such that whenever .

scholarly journals On a conjecture of Z. Jianzhong

1992 ◽  
Vol 45 (1) ◽  
pp. 163-170
Author(s):  
Yasuo Matsugu
Keyword(s):  
Convex Function ◽  
Unit Ball ◽  
Unit Circle ◽  
Orlicz Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Almost All

Let ϕ be a nonnegative, nondecreasing and nonconstant function defined on [0, ∞) such that Φ(t) = ϕ(et) is a convex function on (-∞, ∞). The Hardy-Orlicz space H (ϕ) is defined to be the class of all those functions f holomorphic in the open unit disc of the complex plane C satisfying The subclass H(ϕ)+ of H(ϕ) is defined to be the class of all those functions f ∈ H(ϕ) satisfying for almost all points eit of the unit circle. In 1990, Z. Jianzhong conjectured that H(ϕ)+ = H(ψ)+ if and only if H(ϕ) = H(ψ). In the present paper we prove that it is true not only on the unit disc of C but also on the unit ball of Cn.

scholarly journals Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jorge J. Garcés ◽  
Antonio M. Peralta ◽  
Daniele Puglisi ◽  
María Isabel Ramírez
Keyword(s):  
Unit Ball ◽  
Taylor Series ◽  
Holomorphic Mapping ◽  
Invertible Element ◽  
Holomorphic Mappings ◽  
Open Unit ◽  
Holomorphic Maps ◽  
Open Unit Ball ◽  
C Algebras ◽  
Orthogonally Additive

We study holomorphic maps between C*-algebrasAandB, whenf:BA(0,ϱ)→Bis a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ballU=BA(0,δ). If we assume thatfis orthogonality preserving and orthogonally additive onAsa∩Uandf(U)contains an invertible element inB, then there exist a sequence(hn)inB**and Jordan*-homomorphismsΘ,Θ~:M(A)→B**such thatf(x)=∑n=1∞hnΘ~(an)=∑n=1∞Θ(an)hnuniformly ina∈U. WhenBis abelian, the hypothesis ofBbeing unital andf(U)∩inv(B)≠∅can be relaxed to get the same statement.

scholarly journals On approximating curves associated with nonexpansive mappings

2011 ◽  
Vol 27 (1) ◽  
pp. 142-147
Author(s):  
FRANCESCA VETRO ◽  
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unit Ball ◽  
Nonexpansive Mappings ◽  
Complex Hilbert Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Holomorphic Map ◽  
Nonexpansive Map ◽  
The Hyperbolic Metric

Let X be a Banach space with metric d. Let T, N : X → X be a strict d-contraction and a d-nonexpansive map, respectively. In this paper we investigate the properties of the approximating curve associated with T and N. Moreover, following [3], we consider the approximating curve associated with a holomorphic map f : B → α B and a ρ-nonexpansive map M : B → B, where B is the open unit ball of a complex Hilbert space H, ρ is the hyperbolic metric defined on B and 0 ≤ α < 1. We give conditions on f and M for this curve to be injective, and we show that this curve is continuous.

Asymptotic values of fine continuous functions

1999 ◽  
Vol 127 (1) ◽  
pp. 109-116
Author(s):  
J. R. WORDSWORTH
Keyword(s):  
Continuous Function ◽  
Unit Ball ◽  
Unit Disc ◽  
Polish Space ◽  
Continuous Functions ◽  
Open Unit ◽  
Open Unit Ball ◽  
Analytic Set ◽  
Asymptotic Values ◽  
Complete Separable Metric Space

The set of asymptotic values of a continuous function on the open unit disc in ℝ2 forms an analytic set, in the sense of being a continuous image of a Polish space (complete, separable metric space). This was proved in [9] by J. E. McMillan, who had earlier given versions of this result for holomorphic and meromorphic functions. We extend his method to the case of a function on the open unit ball of ℝn which is continuous merely in the fine topology, the coarsest topology making all subharmonic functions continuous. In particular, we use a version of McMillan's ingenious metric on a certain space of equivalence classes of asymptotic paths. McMillan also proved in [9] that the set of point asymptotic values of a continuous function in the unit disc forms an analytic set. We use a modification of the McMillan metric to extend this result to fine continuous functions in the unit ball and deduce that the set of boundary points of the unit ball at which the function has an asymptotic value forms an analytic set.

Holomorphically embedded discs with rapidly growing area

1999 ◽  
Vol 129 (2) ◽  
pp. 343-349
Author(s):  
Josip Globevnik
Keyword(s):  
Unit Ball ◽  
Open Unit ◽  
Open Unit Ball

It is shown that if V is a closed submanifold of the open unit ball of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ 1. It is also shown that if V is a closed submanifold of ℂ2 biholomorphically equivalent to a disc, then the area of V ∩ r can grow arbitrarily rapidly as r ↗ ∞.

An argument of a function in H1/2

2012 ◽  
Vol 55 (2) ◽  
pp. 507-511
Author(s):  
Takahiko Nakazi ◽  
Takanori Yamamoto
Keyword(s):  
Hardy Space ◽  
Simple Proof ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc

AbstractLet H1/2 be the Hardy space on the open unit disc. For two non-zero functions f and g in H1/2, we study the relation between f and g when f/g ≥ 0 a.e. on ∂D. Then we generalize a theorem of Neuwirth and Newman and Helson and Sarason with a simple proof.

Weak-Star Continuous Analytic Functions

1995 ◽  
Vol 47 (4) ◽  
pp. 673-683 ◽  
Author(s):  
R. M. Aron ◽  
B. J. Cole ◽  
T. W. Gamelin
Keyword(s):  
Unit Ball ◽  
Finite Type ◽  
Analytic Functions ◽  
Approximation Property ◽  
Entire Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Open Unit Ball ◽  
Weakly Continuous ◽  
Weak Star

AbstractLet 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

scholarly journals Harmonic univalent functions defined by post quantum calculus operators

2019 ◽  
Vol 11 (1) ◽  
pp. 5-17 ◽  
Author(s):  
Om P. Ahuja ◽  
Asena Çetinkaya ◽  
V. Ravichandran
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Quantum Calculus ◽  
Special Cases ◽  
The Family ◽  
Covering Theorems

Abstract We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

Some properties of the analytic functions with bounded radius rotation

2019 ◽  
Vol 28 (1) ◽  
pp. 85-90
Author(s):  
YASAR POLATOGLU ◽  
◽  
ASENA CETINKAYA ◽  
OYA MERT ◽  
◽  
...  
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Open Unit ◽  
Coefficient Estimate ◽  
Open Unit Disc ◽  
Radius Of Starlikeness ◽  
Growth Theorem ◽  
Bounded Radius Rotation

In the present paper, we introduce a new subclass of normalized analytic starlike functions by using bounded radius rotation associated with q- analogues in the open unit disc \mathbb D. We investigate growth theorem, radius of starlikeness and coefficient estimate for the new subclass of starlike functions by using bounded radius rotation associated with q- analogues denoted by \mathcal{R}_k(q), where k\geq2, q\in(0,1).

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