Evaluation operators on the Bergman space

1995 ◽  
Vol 117 (3) ◽  
pp. 513-523 ◽  
Author(s):  
Kehe Zhu
Keyword(s):  
Bergman Space ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Space Of Analytic Functions ◽  
Area Measure ◽  
Image Position

Let D be the open unit disc in the complex plane C and let dA be the normalized area measure on D. The Bergman space is the space of analytic functions f in D such that

scholarly journals The Closed Ideals in an Algebra of Analytic Functions

1957 ◽  
Vol 9 ◽  
pp. 426-434 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Banach Algebra ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Complex Functions ◽  
Continuous Complex ◽  
Image Position

Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm (f ∊ ).

scholarly journals The second duals of certain spaces of analytic functions

1970 ◽  
Vol 11 (3) ◽  
pp. 276-280 ◽  
Author(s):  
L. A. Rubel ◽  
A. L. Shields
Keyword(s):  
Banach Space ◽  
Analytic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Closed Subspace ◽  
Open Unit ◽  
Open Unit Disc ◽  
Space Of Analytic Functions ◽  
Image Position ◽  
Spaces Of Analytic Functions

Let ϕ be a continuous, decreasing, real-valued funtion on 0 ≦ r ≦ 1 with ϕ(1) = 0 and ϕ(r) > 0 for r < 1. Let E0 be the Banach space of analytic function f on the open unit disc D, such that f(z)φ(|z|) → 0 as |z| → 1, with norm , where we write ϕ(z) = ϕ(z) for z ∈ D. Let E be the Banach space of analytic functions f on D for which fφ is bounded in D, with the same norm as E0. It is easy to see that E is complete in this norm, and that E0 is a closed subspace of E.

Linear Isometries of some Normed Spaces of Analytic Functions

1985 ◽  
Vol 37 (1) ◽  
pp. 62-74 ◽  
Author(s):  
W. P. Novinger ◽  
D. M. Oberlin
Keyword(s):  
Hardy Space ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Normed Spaces ◽  
Space Of Analytic Functions ◽  
Image Position ◽  
Spaces Of Analytic Functions ◽  
Linear Isometries

For 1 ≦ p < ∞ let Hp denote the familiar Hardy space of analytic functions on the open unit disc D and let ‖·‖ denote the Hp norm. Let Sp denote the space of analytic functions f on D such that f′ ∊ Hp. In this paper we will describe the linear isometries of Sp into itself when Sp is equipped with either of two norms. The first norm we consider is given by(1)and the second by(2)(It is well known [1, Theorem 3.11] that f′ ∊ Hp implies continuity for f on D, the closure of D. Thus (2) actually defines a norm on Sp.) In the former case, with the norm defined by (1), we will show that an isometry of Sp induces, in a sense to be made precise in Section 2, an isometry of Hp and that Forelli's characterization [2] of the isometries of Hp can thus be used to describe the isometries of Hp.

scholarly journals Pointwise bounded approximation by polynomials

1992 ◽  
Vol 112 (1) ◽  
pp. 147-155 ◽  
Author(s):  
Anthony G. O'Farrell ◽  
Fernando Perez-Gonzalez
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Bounded Analytic Functions ◽  
Unbounded Component ◽  
Open Set ◽  
Approximation By Polynomials ◽  
Polynomial Hull ◽  
Image Position

For a bounded open set U ⊂ ℂ, we denote by H∞(U) the collection of all bounded analytic functions on U. We let X denote bdy (U), the boundary of U, Y denote the polynomial hull of U (the complement of the unbounded component of ℂ / X), and U* denote mt (Y), the interior of Y. We denote the sup norm of a function f: A → ℂ by ∥f∥A:We denote the space of all analytic polynomials by ℂ[z], and we denote the open unit disc by D and the unit circle by S1.

Starlike Univalent Functions Bounded on the Real Axis

1989 ◽  
Vol 41 (4) ◽  
pp. 642-658
Author(s):  
Richard Fournier
Keyword(s):  
Real Axis ◽  
Analytic Functions ◽  
Unit Disc ◽  
Univalent Functions ◽  
Uniform Norm ◽  
Open Unit ◽  
Open Unit Disc ◽  
The Real ◽  
Typically Real ◽  
Image Position

We denote by E the open unit disc in C and by H(E) the class of all analytic functions f on E with f(0) = 0. Let (see [3] for more complete definitions)S = {ƒ ∈ H(E)|ƒ is univalent on E}S0 = {ƒ ∈ H(E)|ƒ is starlike univalent on E}TR = {ƒ ∈ H(E)|ƒ is typically real on E}.The uniform norm on (— 1, 1) of a function ƒ ∈ H(E) is defined by

scholarly journals Coefficient inequality for transforms of certain subclass of analytic functions

2017 ◽  
Vol 21 (2) ◽  
pp. 185-193
Author(s):  
T. RamReddy ◽  
D. Shalini ◽  
D. Vamshee Krishna ◽  
B. Venkateswarlu
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Upper Bound ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Toeplitz Determinants ◽  
Coefficient Inequality

The objective of this paper is to obtain the best possible sharp upper bound for the second Hankel functional associated with the kth root transform [f(zk)]1/k of normalized analytic function f(z) when it belongs to certain subclass of analytic functions, defined on the open unit disc in the complex plane using Toeplitz determinants.

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.

Growth of frequently hypercyclic functions for some weighted Taylor shifts on the unit disc

2020 ◽  
pp. 1-18
Author(s):  
Augustin Mouze ◽  
Vincent Munnier
Keyword(s):  
Critical Exponent ◽  
Analytic Functions ◽  
Unit Disc ◽  
Optimal Growth ◽  
Space Of Analytic Functions ◽  
Shift Operators ◽  
Image Position

Abstract For any $\alpha \in \mathbb {R},$ we consider the weighted Taylor shift operators $T_{\alpha }$ acting on the space of analytic functions in the unit disc given by $T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$ $ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^{k}.\end{align*}$ We establish the optimal growth of frequently hypercyclic functions for $T_\alpha $ in terms of $L^p$ averages, $1\leq p\leq +\infty $ . This allows us to highlight a critical exponent.

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