scholarly journals Radial growth and exceptional sets for Cauchy–Stieltjes integrals

1994 ◽  
Vol 37 (1) ◽  
pp. 73-89 ◽  
Author(s):  
D. J. Hallenbeck ◽  
T. H. MacGregor
Keyword(s):  
Unit Circle ◽  
Radial Growth ◽  
Borel Measure ◽  
Local Conditions ◽  
Exceptional Sets ◽  
Image Position ◽  
Complex Valued

This paper considers the radial and nontangential growth of a function f given bywhere α>0 and μ is a complex-valued Borel measure on the unit circle. The main theorem shows how certain local conditions on μ near eiθ affect the growth of f(z) as z→eiθ in Stolz angles. This result leads to estimates on the nontangential growth of f where exceptional sets occur having zero β-capacity.

The Isometries of Hp

1964 ◽  
Vol 16 ◽  
pp. 721-728 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Circle ◽  
Lebesgue Measure ◽  
Measurable Functions ◽  
Image Position ◽  
Complex Valued

Let a be the Lebesgue measure on the unit circle |z| = 1 withand let Lp be the space of complex-valued σ-measurable functions f such thatis finite. Hp is the closure in Lp of the algebra of analytic polynomials

scholarly journals Extreme points in spaces between Dirichlet and Vanishing Mean Oscillation

2003 ◽  
Vol 67 (3) ◽  
pp. 365-375
Author(s):  
K. J. Wirths ◽  
J. Xiao
Keyword(s):  
Unit Ball ◽  
Unit Circle ◽  
Dirichlet Space ◽  
Extreme Points ◽  
Closed Unit Ball ◽  
Mean Oscillation ◽  
Image Position ◽  
Complex Valued

For p ∈ (0, ∞) define Qp0(∂Δ) as the space of all Lebesgue measurable complex-valued functions f; on the unit circle ∂Δ for which ∫∂Δf;(z)|dz|/(2π) = 0 andas the open subarc I of ∂Δ varies. Note that each Qp,0(∂Δ) lies between the Dirichlet space and Sarason's vanishing mean oscillation space. This paper determines the extreme points of the closed unit ball of Qp,0(∂Δ) equipped with an appropriate norm.

Mean values and classes of harmonic functions

1984 ◽  
Vol 96 (3) ◽  
pp. 501-505 ◽  
Author(s):  
Thomas Ramsey ◽  
Yitzhak Weit
Keyword(s):  
Unit Circle ◽  
Harmonic Functions ◽  
Borel Measure ◽  
Mean Value ◽  
Finite Complex ◽  
Mean Values ◽  
Complex Borel Measure ◽  
Generalized Mean ◽  
Image Position

Let μ be a finite complex Borel measure supported on the unit circle.In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form.and for all ξ ∈ , | ξ | = R for some fixed R > 0.

scholarly journals Omitted rays and wedges of fractional Cauchy transforms

2006 ◽  
Vol 80 (3) ◽  
pp. 367-373
Author(s):  
R. A. Hibschweiler ◽  
T. H. Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Borel Measure ◽  
Cauchy Transforms ◽  
Complex Valued

AbstractFor α > 0 let α denote the set of functions which can be expressed where μ is a complex-valued Borel measure on the unit circle. We show that if f is an analytic function in Δ = {z ∈ : |z| < 1} and there are two nonparallel rays in /f(Δ) which do not meet, then f ∈ α where απ denotes the largest of the two angles determined by the rays. Also if the range of a function analytic in Δ is contained in an angular wedge of opening απ and 1 < α < 2, then f ∈ α.

scholarly journals A Note on the Kawada-into Theorem

1963 ◽  
Vol 13 (4) ◽  
pp. 295-296 ◽  
Author(s):  
John S. Pym
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Regular Borel Measure ◽  
Image Position ◽  
Complex Valued

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).

Functions regular in the unit circle

1956 ◽  
Vol 52 (1) ◽  
pp. 49-60 ◽  
Author(s):  
C. N. Linden ◽  
M. L. Cartwright
Keyword(s):  
Unit Circle ◽  
Positive Constant ◽  
Upper Bounds ◽  
Image Position

Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

scholarly journals Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

1962 ◽  
Vol 14 ◽  
pp. 540-551 ◽  
Author(s):  
W. C. Royster
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Simple Pole ◽  
Image Position ◽  
Inverse Functions ◽  
Variational Formulas

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

p-Radial Exceptional Sets and Conformal Mappings

2007 ◽  
Vol 50 (4) ◽  
pp. 579-587
Author(s):  
Piotr Kot
Keyword(s):  
Holomorphic Function ◽  
Unit Disc ◽  
Conformal Mappings ◽  
Exceptional Sets ◽  
Image Position

AbstractFor p > 0 and for a given set E of type Gδ in the boundary of the unit disc ∂ we construct a holomorphic function f ∈ such thatIn particular if a set E has a measure equal to zero, then a function f is constructed as integrable with power p on the unit disc .

Multipliers of Fractional Cauchy Transforms and Smoothness Conditions

1998 ◽  
Vol 50 (3) ◽  
pp. 595-604 ◽  
Author(s):  
Donghan Luo ◽  
Thomas Macgregor
Keyword(s):  
Analytic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Borel Measure ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cauchy Transforms ◽  
The Family ◽  
Complex Borel Measure

AbstractThis paper studies conditions on an analytic function that imply it belongs to Mα, the set of multipliers of the family of functions given by where μ is a complex Borel measure on the unit circle and α > 0. There are two main theorems. The first asserts that if 0 < α < 1 and sup. The second asserts that if 0 < α < 1, ƒ ∈ H∞ and supt. The conditions in these theorems are shown to relate to a number of smoothness conditions on the unit circle for a function analytic in the open unit disk and continuous in its closure.

Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

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