scholarly journals Domination of the supremum of a bounded harmonic function by its supremum over a countable subset

1987 ◽  
Vol 30 (3) ◽  
pp. 471-477 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Harmonic Function ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Open Unit ◽  
Countable Subset ◽  
Open Unit Disc ◽  
Image Position ◽  
Bounded Harmonic Functions ◽  
Bounded Harmonic Function

For what sequences {an} of points of the open unit disc D does there exist a constant k such thatfor all bounded harmonic functions f on D?

scholarly journals A series expansion for generalized harmonic functions

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Markus Klintborg ◽  
Anders Olofsson
Keyword(s):  
Harmonic Function ◽  
Series Expansion ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Poisson Kernel ◽  
Open Unit ◽  
Open Unit Disc ◽  
Generalized Harmonic Function ◽  
Generalized Harmonic Functions

AbstractWe consider a class of generalized harmonic functions in the open unit disc in the complex plane. Our main results concern a canonical series expansion for such functions. Of particular interest is a certain individual generalized harmonic function which suitably normalized plays the role of an associated Poisson kernel.

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.

6.—Rationally Convex Hulls and Potential Theory.

1976 ◽  
Vol 74 ◽  
pp. 81-89
Author(s):  
Richard F. Basener
Keyword(s):  
Potential Theory ◽  
Compact Subset ◽  
Unit Disc ◽  
Rational Functions ◽  
Continuous Functions ◽  
Uniform Algebra ◽  
Convex Hulls ◽  
Open Unit ◽  
Open Unit Disc ◽  
Image Position

SynopsisLet S be a compact subset of the open unit disc in C. Associate to S the setLet R(X) be the uniform algebra on X generated by the rational functions which are holomorphic near X. It is shown that the spectrum of R(X) is determined in a simple wayby the potential-theoretic properties of S. In particular, the spectrum of R(X) is X if and only if the functions harmonic near S are uniformly dense in the continuous functions on S. Similar results can be obtained for other subsets of C2 constructed from compact subsets of C.

scholarly journals Mapping properties of Janowski-type harmonic functions involving Mittag-Leffler function

2021 ◽  
Vol 6 (12) ◽  
pp. 13235-13246
Author(s):  
Murugusundaramoorthy Gangadharan ◽  
◽  
Vijaya Kaliyappan ◽  
Hijaz Ahmad ◽  
K. H. Mahmoud ◽  
...  
Keyword(s):  
Convolution Operator ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disc

<abstract><p>In this paper, we examine a connotation between certain subclasses of harmonic univalent functions by applying certain convolution operator regarding Mittag-Leffler function. To be more precise, we confer such influences with Janowski-type harmonic univalent functions in the open unit disc $ \mathbb{D}. $</p></abstract>

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.

scholarly journals Convergence criteria for bounded sequences

1972 ◽  
Vol 18 (2) ◽  
pp. 99-103 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Unit Disc ◽  
Open Unit ◽  
Complex Numbers ◽  
Convergence Criteria ◽  
Open Unit Disc ◽  
Primary Object ◽  
Image Position ◽  
Bounded Sequences

Let {Kn} be a sequence of complex numbers, letand letLet D be the open unit disc {z: |z| <1}, let be its closure and let .The primary object of this paper is to prove the two theorems stated below, the first of which generalises a result of Copson (1).

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 Vanishing l1-sums of the Poisson kernel, and sums with positive coefficients

1989 ◽  
Vol 32 (3) ◽  
pp. 431-447 ◽  
Author(s):  
F. F. Bonsall ◽  
D. Walsh
Keyword(s):  
Unit Disc ◽  
Poisson Kernel ◽  
Open Unit ◽  
Complex Numbers ◽  
Open Unit Disc ◽  
Image Position ◽  
Positive Coefficients

For z in D and ζ in ∂D, we denote by pz(ζ) the Poisson kernel (1 − │z│2)│1 − z̄ζ−2 for the open unit disc D. We ask for what countable sets {an:n∈ℕ} of points of D there exist complex numbers λn withby which we mean that the series converges to zero in the norm of L1(∂D).

Inequalities for the derivatives of a bounded harmonic function

1948 ◽  
Vol 44 (2) ◽  
pp. 155-158 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Unit Circle ◽  
Harmonic Functions ◽  
Conformal Transformations ◽  
Alternative Method ◽  
Schwarz’S Lemma ◽  
The One ◽  
Image Position ◽  
Derivatives Of ◽  
Bounded Harmonic Function

If h(r, θ) is harmonic in the unit circle | r | < 1 and satisfies the condition | h | ≤ 1, then there is a function u(ø) which satisfies | u | ≤ 1 such thatand conversely. Hence, any properties of such harmonic functions should be deducible from equation (1). A number of such properties have been proved by Koebe (Math. Z. 6 (1920), 52–84, 69), using Schwarz's lemma and the geometry of simple conformal transformations. They can be deduced from (1) together with an elementary lemma on the rearrangement of a function (Lemma 1 below). As, however, students of this subject will regard Koebe's method as the one best adapted to establish his theorems, we shall illustrate the alternative method by considering two new problems, namely to find max ∂h/∂r, max ∂h/∂θ, where the maximum in each case is taken for all harmonic functions h which satisfy

scholarly journals Properties of harmonic functions which are convex of order $ \bf \beta $ with respect to symmetric points

2009 ◽  
Vol 40 (1) ◽  
pp. 31-39 ◽  
Author(s):  
Aini Janteng ◽  
Suzeini Abdul Halim
Keyword(s):  
Unit Disc ◽  
Harmonic Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Symmetric Points ◽  
Complex Valued ◽  
Class Of Functions

Let $ \mathcal{H} $ denote the class of functions $ f $ which are harmonic and univalent in the open unit disc $ {D=\{z:|z|<1\}} $. This paper defines and investigates a family of complex-valued harmonic functions that are orientation preserving and univalent in $ \mathcal{D} $ and are related to the functions convex of order $ \beta(0\leq \beta <1) $, with respect to symmetric points. We obtain coefficient conditions, growth result, extreme points, convolution and convex combinations for the above harmonic functions.

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