Thomas H. MacGregor
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View article: Derivatives of the Hyperbolic Density Near an Isolated Boundary Point
Derivatives of the Hyperbolic Density Near an Isolated Boundary Point Open
Suppose that c is an isolated boundary point of a hyperbolic domain Ω in the complex plane, and let λ Ω denote the density of the hyperbolic metric on Ω.We show that for each pair of nonnegative integers n and mwhere c 0 = 1 andfor n = 1, …
View article: Omitted rays and wedges of fractional Cauchy transforms
Omitted rays and wedges of fractional Cauchy transforms Open
For 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)…
View article: Multipliers of Fractional Cauchy Transforms and Smoothness Conditions
Multipliers of Fractional Cauchy Transforms and Smoothness Conditions Open
This paper studies conditions on an analytic function that imply it belongs to M α , 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. T…
View article: Conditions on the Logarithmic Derivative of a Function Implying Boundedness
Conditions on the Logarithmic Derivative of a Function Implying Boundedness Open
In this paper we investigate functions analytic and nonvanishing in the unit disk, with the property that the logarithmic derivative is contained in some domain $\Omega$. We obtain conditions on $\Omega$ which imply that the functions are …
View article: Radial Limits and growths of fractional Cauchy transforms
Radial Limits and growths of fractional Cauchy transforms Open
for |z| < 1. Fα is a Banach space with respect to the norm defined by ‖f‖Fα = inf ‖μ‖ where μ varies over all measures in M for which (1) or (2) holds and where ‖μ‖ is the total variation norm of μ. The spaces Fα have been studied in a num…
View article: Radial growth and exceptional sets for Cauchy–Stieltjes integrals
Radial growth and exceptional sets for Cauchy–Stieltjes integrals Open
This paper considers the radial and nontangential growth of a function f given by where 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 e i θ affect the growth o…
View article: Multipliers of Families of Cauchy-Stieltjes Transforms
Multipliers of Families of Cauchy-Stieltjes Transforms Open
For $\alpha > 0$ let ${\mathcal {F}_\alpha }$ denote the class of functions defined for $|z| < 1$ by integrating $1/{(1 - xz)^\alpha }$ against a complex measure on $|x|= 1$. A function $g$ holomorphic in $|z| < 1$ is a multiplier of ${\ma…
View article: Closure Properties of Families of Cauchy-Stieltjes Transforms
Closure Properties of Families of Cauchy-Stieltjes Transforms Open
For a > 0 let ^ denote the class of functions defined for \z\ < 1 by integrating 1/(1 -xz)" against a complex measure on \x\ = 1 .The main results in this paper assert that !7a is closed under multiplication by a function holomorphic for \…
View article: Radial growth and variation of bounded analytic functions
Radial growth and variation of bounded analytic functions Open
If a function f analytic in Δ = { z ∈ℂ:| z |<1} has a nontangential limit as z → e i θ , then lim r →1− (1− r ) f ′( re i θ )=0 [7, p. 181). It follows that this limit is zero for almost all θ for a number of classes of functions including…
View article: Support points of families of analytic functions described by subordination
Support points of families of analytic functions described by subordination Open
We determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination. The families we study include the functions with a positive real part, the t…
View article: Families of Real and Symmetric Analytic Functions
Families of Real and Symmetric Analytic Functions Open
We introduce families of functions analytic in the unit disk and having rotational symmetries. The families include the $k$-fold symmetric univalent functions which have real coefficients. We relate the families to special classes of funct…
View article: Variability Regions for Bounded Analytic Functions with Applications to Families Defined by Subordination
Variability Regions for Bounded Analytic Functions with Applications to Families Defined by Subordination Open
We examine the set of points $(\varphi (\zeta ),\varphi â(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))$ where $|\zeta | < 1$ and $\varphi$ varies over the class of functions analytic in the open unit disk and is either (1) uniformly bounde…
View article: Matrix transformation of univalent power series
Matrix transformation of univalent power series Open
Suppose that A = [α nk ], ( n, k = 0, 1, 2, …), is an indinite matrix with complex entries. A transforms a complex sequence a = { a n to a complex sequence { b n = b = Aa where assuming that the series in (1) converges. Each sequence a = {…
View article: Convex Hulls and Extreme Points of Families of Starlike and Convex Mappings
Convex Hulls and Extreme Points of Families of Starlike and Convex Mappings Open
The closed convex hull and extreme points are obtained for the starlike functions of order $\alpha$ and for the convex functions of order $\alpha$. More generally, this is determined for functions which are also k-fold symmetric. Integral …
View article: Hull Subordination and Extremal Problems for Starlike and Spirallike Mappings
Hull Subordination and Extremal Problems for Starlike and Spirallike Mappings Open
Let $\mathfrak {F}$ be a compact subset of the family $\mathcal {A}$ of functions analytic in $\Delta = \{ z:\;|z| < 1\}$, and let $\mathcal {L}$ be a continuous linear operator of order zero on $\mathcal {A}$. We show that if the extreme …
View article: Approximation by Polynomials Subordinate to a Univalent Function
Approximation by Polynomials Subordinate to a Univalent Function Open
This paper is concerned with approximating a function /(z) analytic and univalent in the unit disk £={z: \z\ < 1} by polynomials which are also univalent in £.We are interested in such approximations where exactly one polynomial of each de…
View article: An Inequality Concerning Analytic Functions with a Positive Real Part
An Inequality Concerning Analytic Functions with a Positive Real Part Open
This paper contains an inequality about functions which are analytic and have a positive real part in the unit disk. A first consequence of the inequality is the fact that if is analytic for | z | < 1 and has values lying in a strip of wid…
View article: A class of univalent functions
A class of univalent functions Open
f'(0)= 1. In this paper we study the subclass denoted by F and defined by the condition If'(z) - Ij < 1 for I zj < 1. Some of our results have already been proven for the particular functions in F whose
View article: The radius of convexity for starlike functions of order 1\over2
The radius of convexity for starlike functions of order 1\over2 Open
As Strohhicker has pointed out neither (3) nor (4) are sufficient for the convexity of f(z) in Ilz <1. Indeed, a function may satisfy (4) without being univalent for I zj < 1. In this paper we determine the largest circle I zj <r such that…
View article: Functions Whose Derivative has a Positive Real Part
Functions Whose Derivative has a Positive Real Part Open
Introduction.Let P denote the class of functions which are regular and satisfy Re /' (z) > 0 for | z \ < 1 and are normalized by /(0) = 0 and /' (0) = 1.This paper develops some properties of functions in P.An early consideration of functi…