On families of Cauchy transforms and BMOA (original) (raw)
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Cauchy Transforms and Univalent Functions
Fields Institute Communications, 2013
We use a formula of Pommerenke relating the primitives of functions which are the Cauchy transforms of measures on the unit circle to their behavior in the space of functions of bounded mean oscillation. This is a linear process and it has some smoothness. Further, there is a non-linear map from the Cauchy transforms into the normalized univalent functions. We show that for the subspace H 1 of Cauchy transforms the univalent functions so obtained have quasi-conformal extensions to all of the plane.
Absolutely continuous measures and compact composition operator on spaces of Cauchy transforms
International Journal of Mathematics and Mathematical Sciences, 2004
The analytic self-map of the unit diskD,φis said to induce a composition operatorCφfrom the Banach spaceXto the Banach spaceYifCφ(f)=f∘φ∈Yfor allf∈X. Forz∈Dandα>0, the families of weighted Cauchy transformsFαare defined byf(z)=∫TKxα(z)dμ(x), whereμ(x)is complex Borel measure,xbelongs to the unit circleT, and the kernelKx(z)=(1−x¯z)−1. In this paper, we will explore the relationship between the compactness of the composition operatorCφacting onFαand the complex Borel measuresμ(x).
Journal of Interpolation and Approximation in Scientific Computing, 2012
In this article, we introduce a new class of analytic functions of the unit disc D namely the Exponential Cauchy Transforms K e defined by f (z) = ∫ T exp [K (xz)] dµ(x) where K (z) = (1 − z) −1 is classical Cauchy kernel and µ(x) is a complex Borel measures and x belongs to the unit circle T. We use Laguerre polynomials to explore the coefficients of the Taylor expansions of the kernel and Peron's formula to study the asymptotic behavior of the Taylor coefficients. Finally we investigate relationships between our new class K e , the classical Cauchy space K and the Hardy spaces H p .
Some Generalized Convolution Properties Associated With Certain Subclasses of Analytic Functions
J. Inequal. Pure Appl. Math, 2002
For functions belonging to each of the subclasses M * n (α) and N * n (α) of normalized analytic functions in open unit disk U, which are introduced and investigated in this paper, the authors derive several properties involving their generalized convolution by applying certain techniques based especially upon the Cauchy-Schwarz and Hölder inequalities. A number of interesting consequences of these generalized convolution properties are also considered.
Certain Transformations Preserving Families of Univalent Analytic Functions
2015
The article deals with the family mathcalU(lambda){\mathcal U}(\lambda)mathcalU(lambda) of all functions fff normalized and analytic in the unit disk such that big∣big(z/f(z)big)2f′(z)−1big∣<lambda\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda big∣big(z/f(z)big)2f′(z)−1big∣<lambda for some 0<lambdaleq10<\lambda \leq 10<lambdaleq1. The family mathcalU(lambda){\mathcal U}(\lambda)mathcalU(lambda) has been studied extensively in the recent past and functions in this family are known to be univalent in ID\IDID. However, the problem of determining sharp bounds for the second coefficients of functions in this family was solved recently in \cite{VY2013} by Vasudevarao and Yanagihara but the proof was complicated. In this article, we first present a simpler proof. We obtain a number of new subordination results for this family and their consequences. In addition, we show that the family mathcalU(lambda){\mathcal U}(\lambda )mathcalU(lambda) is preserved under a number of elementary transformations such as rotation, conjugation, dilation and omitted value transformations, but surprisingly this family is not preserved under the nnn-th root transformation for any ngeq2n\geq 2ngeq2. This is a basic here which helps to generate a number of new theorems and in particular provides a way for constructions of functions from the family mathcalU(lambda){\mathcal U}(\lambda)mathcalU(lambda). Finally, we deal with a radius problem.
On an Integral Transform of a Class of Analytic Functions
Abstract and Applied Analysis, 2012
For α, γ ≥ 0 and β < 1, let W β α, γ denote the class of all normalized analytic functions f in the open unit disc E {z : |z| < 1} such that e iφ 1 − α 2γ f z /z α − 2γ f z γzf z − β > 0, z ∈ E for some φ ∈ R. It is known Noshiro 1934 and Warschawski 1935 that functions in W β 1, 0 are close-to-convex and hence univalent for 0 ≤ β < 1. For f ∈ W β α, γ , we consider the integral transform F z V λ f z : 1 0 λ t f tz /t dt, where λ is a nonnegative real-valued integrable function satisfying the condition 1 0 λ t dt 1. The aim of present paper is, for given δ < 1, to find sharp values of β such that i V λ f ∈ W δ 1, 0 whenever f ∈ W β α, γ and ii V λ f ∈ W δ α, γ whenever f ∈ W β α, γ .
European Journal of Pure and Applied Mathematics, 2010
The purpose of the present paper is to derive some inclusion properties and argument estimates of certain normalized analytic functions in the open unit disk, which are defined by means of a class of multiplier transformations. Furthermore, the integral preserving properties in a sector are investigated for these multiplier transformations. 2000 Mathematics Subject Classifications: 30C45
Cauchy transforms of measures viewed as some functionals of Fourier transforms
In memory of Kazimierz Urbanik ABSTRACT. The Cauchy transform of a positive measure plays an important role in complex analysis and more recently in so-called free probability. We show here that the Cauchy transform restricted to the imaginary axis can be viewed as the Fourier transform of some corresponding measures. Thus this allows the full use of that classical tool. Furthermore, we relate restricted Cauchy transforms to classical compound Poisson measures, exponential mixtures, geometric infinite divisibility and free-infinite divisibility. Finally we illustrate our approach with some examples.