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Exponential Cauchy Transforms

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Author(s): El-Bachir Yallaoui

Journal: Journal of Interpolation and Approximation in Scientific Computing
ISSN 2194-3907

Volume: 2012;
Start page: 1;
Date: 2012;
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Keywords: Exponential Cauchy transforms | Laguerre polynomial | Turan's inequality

ABSTRACT
In this article, we introduce a new class of analytic functions of the unit disc $mathbf{D}$ namely the Exponential Cauchy Transforms $mathbf{{K}_{e}}$ defined by f(z)= {displaystyleint_{mathbf{T}}} expleft[ Kleft( xzight) ight] dmu(x) where $Kleft( zight) =left( 1-zight) ^{-1}$ is classical Cauchy kernel and $mu(x)$ is a complex Borel measures and $x$ belongs to the unit circle $mathbf{T}$ . 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 $mathbf{{K}_{e}}$, the classical Cauchy space $mathbf{K}$ and the Hardy spaces $H^{p}$.
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