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On the radius of convexity for a class of conformal maps

100%
Karunakaran V., Bhuvaneswari K.
Colloquium Mathematicum
|
2007
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tom 109
|
nr 2
251-256
EN
Let 𝓐 denote the class of all analytic functions f in the open unit disc 𝔻 in the complex plane satisfying f(0) = 0, f'(0) = 1. Let U(λ) (0 < λ ≤ 1) denote the class of functions f ∈ 𝓐 for which |(z/f(z))²f'(z) -1| < λ for z ∈ 𝔻. The behaviour of functions in this class has been extensively studied in the literature. In this paper, we shall prove that no member of U₀(λ) = {f ∈ U(λ): f''(0) = 0} is convex in 𝔻 for any λ and obtain a lower bound for the radius of convexity for the family U₀(λ). These results settle a conjecture proposed in the literature negatively. We also improve the existing lower bound for the radius of convexity of the family U₀(λ).
2
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Gelfand transform for a Boehmian space of analytic functions

100%
Karunakaran V., Chella Rajathi R.
Annales Polonici Mathematici
|
2011
|
tom 101
|
nr 1
39-45
EN
Let $H^∞(𝔻)$ denote the usual commutative Banach algebra of bounded analytic functions on the open unit disc 𝔻 of the finite complex plane, under Hadamard product of power series. We construct a Boehmian space which includes the Banach algebra A where A is the commutative Banach algebra with unit containing $H^∞(𝔻)$. The Gelfand transform theory is extended to this setup along with the usual classical properties. The image is also a Boehmian space which includes the Banach algebra C(Δ) of continuous functions on the maximal ideal space Δ (where Δ is given the usual Gelfand topology). It is shown that every F ∈ C(Δ) is the Gelfand transform of a suitable Boehmian. It should be noted that in the classical theory the Gelfand transform from A into C(Δ) is not surjective even though it can be shown that the image is dense. Thus the context of Boehmians enables us to identify every element of C(Δ) as the Gelfand transform of a suitable convolution quotient of analytic functions. (Here the convolution is the Hadamard convolution).
3
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The Laplace transform on a Boehmian space

100%
Karunakaran V., Prasanna Devi C.
Annales Polonici Mathematici
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2010
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tom 97
|
nr 2
151-157
EN
In the literature a Boehmian space containing all right-sided Laplace transformable distributions is defined and studied. Besides obtaining basic properties of this Laplace transform, an inversion formula is also obtained. In this paper we shall improve upon two theorems one of which relates to the continuity of this Laplace transform and the other is concerned with the inversion formula.
4
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Boehmians of type S and their Fourier transforms

100%
Bhuvaneswari R., Karunakaran V.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2010
|
tom 64
|
nr 1
EN
Function spaces of type S are introduced and investigated in the literature. They are also applied to study the Cauchy problem. In this paper we shall extend the concept of these spaces to the context of Boehmian spaces and study the Fourier transform theory on these spaces. These spaces enable us to combine the theory of Fourier transform on these function spaces as well as their dual spaces.
5
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Operational calculus and Fourier transform on Boehmians

100%
Karunakaran V., Roopkumar R.
Colloquium Mathematicum
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2005
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tom 102
|
nr 1
21-32
EN
We define various operations on the space of ultra Boehmians like multiplication with certain analytic functions which are Fourier transforms of compactly supported distributions, polynomials, and characters $(e^{ist}, s,t ∈ ℝ)$, translation, differentiation. We also prove that the Fourier transform on the space of ultra Boehmians has all the operational properties as in the classical theory.
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