Averages of holomorphic mappings

1986 ◽  
Vol 100 (2) ◽  
pp. 371-381 ◽  
Author(s):  
V. Nestoridis
Keyword(s):  
Unit Circle ◽  
Lebesgue Measure ◽  
Unit Disc ◽  
Holomorphic Mappings ◽  
Open Unit ◽  
Open Unit Disc ◽  
Disc Algebra ◽  
Several Variables

In this paper we present two versions in several variables of the following result:Theorem 1([2, 3]). Let f be a function in the disc algebra (more generally in H1). Then for every point z0 in the open unit disc, there is an interval I on the unit circle T such that f(z0) = 1/|I| ∫Ifdσ, where 0 < |I| ≤ 2π denotes the length of I and σ the Lebesgue measure on T.

The Essential Spectrum of the Essentially Isometric Operator

2014 ◽  
Vol 57 (1) ◽  
pp. 145-158
Author(s):  
H. S. Mustafayev
Keyword(s):  
Hilbert Space ◽  
Unit Circle ◽  
Essential Spectrum ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Isometric Operator ◽  
Disc Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

AbstractLet T be a contraction on a complex, separable, infinite dimensional Hilbert space and let σ(T) (resp. σe(T)) be its spectrum (resp. essential spectrum). We assume that T is an essentially isometric operator; that is, IH -T*T is compact. We show that if D\σ(T) ≠ Ø, then for every f from the disc-algebraσe(f(T)) = f(σe(T))where D is the open unit disc. In addition, if T lies in the class C0.∪ C.0, thenσe(f(T)) = f(σ(T) ∩ Γ),where Γ is the unit circle. Some related problems are also discussed.

scholarly journals On the Asymptotic Behavior of Functions Harmonic in a Disc

1966 ◽  
Vol 28 ◽  
pp. 187-191
Author(s):  
J. E. Mcmillan
Keyword(s):  
Asymptotic Behavior ◽  
Unit Circle ◽  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Continuous Curve ◽  
Open Unit Disc ◽  
Boundary Path

Let D be the open unit disc, and let C be the unit circle in the complex plane. Let f be a (real-valued) function that is harmonic in D. A simple continuous curve β: z(t) (0≦t<1) contained in D such that |z(t)|→1 as t→1 is a boundary path with end (the bar denotes closure).

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.

On the Zeros of Functions with Derivatives in H1 and H∞

1970 ◽  
Vol 22 (2) ◽  
pp. 342-347 ◽  
Author(s):  
James Wells
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Open Unit ◽  
Hardy Class ◽  
Open Unit Disc ◽  
Zeros Of Functions ◽  
Limit Points ◽  
Image Position ◽  
Derivatives Of

Let {zk},0 < |zk| < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk} which will ensure the existence of a function f analytic in D satisfying(A)and whose derivative f′ belongs to the Hardy class H1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.THEOREM 1. If(1)(2)and(3)then there is a function f analytic in D which satisfies (A) and its derivative f′ belongs to H1.

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.

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).

Approximation by Unimodular Functions

1971 ◽  
Vol 23 (2) ◽  
pp. 257-269 ◽  
Author(s):  
Stephen Fisher
Keyword(s):  
Blaschke Product ◽  
Uniform Approximation ◽  
Unit Disc ◽  
Open Unit ◽  
Blaschke Products ◽  
Open Unit Disc ◽  
Pointwise Approximation ◽  
Finite Blaschke Product ◽  
Image Position ◽  
Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

Coefficient Inequalities for Lp-Valued Analytic Functions

1981 ◽  
Vol 24 (3) ◽  
pp. 347-350
Author(s):  
Lawrence A. Harris
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Coefficient Inequalities

AbstractA Hausdorff-Young theorem is given for Lp-valued analytic functions on the open unit disc and estimates on such functions and their derivatives are deduced.

scholarly journals On Certain Subclass of Harmonic Starlike Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. Y. Lashin
Keyword(s):  
Integral Operator ◽  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Starlike Functions ◽  
Open Unit ◽  
Open Unit Disc ◽  
New Class ◽  
Distortion Bounds ◽  
Harmonic Starlike

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.

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