The Experts below are selected from a list of 297 Experts worldwide ranked by ideXlab platform
Vladimir V Andrievskii - One of the best experts on this subject based on the ideXlab platform.
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Bernstein Polynomial Inequality on a Compact Subset of the Real Line
arXiv: Complex Variables, 2018Co-Authors: Vladimir V AndrievskiiAbstract:We prove an analogue of the classical Bernstein polynomial inequality on a compact subset $E$ of the real line. The Lipschitz continuity of the Green function for the complement of $E$ with respect to the Extended Complex Plane and the differentiability at a point of $E$ of a special, associated with $E$, conformal mapping of the upper half-Plane onto the comb domain play crucial role in our investigation.
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On Sparse Sets with the Green Function of the Highest Smoothness
Computational Methods and Function Theory, 2006Co-Authors: Vladimir V AndrievskiiAbstract:Let E be a regular compact subset of the real line, let be the Green function of the complement of E with respect to the Extended Complex Plane ${\overline {\rm C}}$ with pole at ∞. We construct two examples of sets E of the minimum Hausdorff dimension with satisfying the Hölder condition with p = 1/2 either uniformly or locally.
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On Sparse Sets with the Green Function of the Highest Smoothness
Computational Methods and Function Theory, 2006Co-Authors: Vladimir V AndrievskiiAbstract:Let E be a regular compact subset of the real line, let Open image in new window be the Green function of the complement of E with respect to the Extended Complex Plane \({\overline {\rm C}}\) with pole at ∞. We construct two examples of sets E of the minimum Hausdorff dimension with Open image in new window satisfying the Holder condition with p = 1/2 either uniformly or locally.
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the highest smoothness of the green function implies the highest density of a set
Arkiv för Matematik, 2004Co-Authors: Vladimir V AndrievskiiAbstract:We investigate local properties of the Green function of the complement of a compact setEυ[0,1] with respect to the Extended Complex Plane. We demonstrate, that if the Green function satisfies the 1/2-Holder condition locally at the origin, then the density ofE at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]..
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Uniformly Perfect Subsets of the Real Line and John Domains
Computational Methods and Function Theory, 2004Co-Authors: Vladimir V AndrievskiiAbstract:Let E be a regular compact subset of the real line. We relate the Green function for the complement of E (with respect to the Extended Complex Plane) to some conformal map f. We establish the relationship between the geometry of E and f(E).
Goutam Satpati - One of the best experts on this subject based on the ideXlab platform.
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Loewner chain and quasiconformal extension of some classes of univalent functions
Complex Variables and Elliptic Equations, 2019Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the Extended Complex Plane. Some of these extens...
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On some results for a class of meromorphic functions having quasiconformal extension
Proceedings - Mathematical Sciences, 2018Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:We consider the class \(\Sigma (p)\) of univalent meromorphic functions f on \({\mathbb D}\) having a simple pole at \(z=p\in [0,1)\) with residue 1. Let \(\Sigma _k(p)\) be the class of functions in \(\Sigma (p)\) which have k-quasiconformal extension to the Extended Complex Plane \({\hat{\mathbb C}}\), where \(0\le k < 1\). We first give a representation formula for functions in this class and using this formula, we derive an asymptotic estimate of the Laurent coefficients for the functions in the class \(\Sigma _k(p)\). Thereafter, we give a sufficient condition for functions in \(\Sigma (p)\) to belong to the class \(\Sigma _k(p).\) Finally, we obtain a sharp distortion result for functions in \(\Sigma (p)\) and as a consequence, we obtain a distortion estimate for functions in \(\Sigma _k(p).\)
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Loewner chain and quasiconformal extension of some classes of univalent functions
arXiv: Complex Variables, 2018Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the Extended Complex Plane. Some of these extensions have been obtained by constructing suitable Loewner chains and others have been obtained by applying a well-known result.
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Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension
2017Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:Let $\Sigma_k(p)$ be the class of univalent meromorphic functions defined on $\mathbb{D}$ with $k$-quasiconformal extension to the Extended Complex Plane $\widehat{\mathbb{C}}$, where $0\leq k < 1$. Let $\Sigma_k^0(p)$ be the class of functions $f \in \Sigma_k(p)$ having expansion of the form $f(z)= 1/(z-p) + \sum_{n=1}^{\infty}b_n z^{n}$ on $\mathbb{D}$. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in $\Sigma_k^0(p)$. As a consequence of the obtained results, we present a sharp estimate for the bounds of the Hilbert transform.
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On some results for meromorphic univalent functions having quasiconformal extension
arXiv: Complex Variables, 2017Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:We consider the class $\Sigma(p)$ of univalent meromorphic functions $f$ on $\ID$ having simple pole at $z=p\in[0,1)$ with residue 1. Let $\Sigma_k(p)$ be the class of functions in $\Sigma(p)$ which have $k$-quasiconformal extension to the Extended Complex Plane $\sphere$ %with $q=\frac{1+k}{1-k}$ where $0\leq k < 1$. We first give a representation formula for functions in this class and using this formula we derive an asymptotic estimate of the Laurent coefficients for the functions in the class $\Sigma_k(p)$. Thereafter we give a sufficient condition for functions in $\Sigma(p)$ to belong in the class $\Sigma_k(p).$ Finally we obtain a sharp distortion result for functions in $\Sigma(p)$ and as a consequence, we get a distortion estimate for functions in $\Sigma_k(p).$
Soonchil Lee - One of the best experts on this subject based on the ideXlab platform.
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Qubit Geometry and Conformal Mapping
Quantum Information Processing, 2002Co-Authors: Janice C. Lee, Jaewan Kim, Chang-ho Kim, E. K. Lee, Soonchil LeeAbstract:Identifying the Bloch sphere with the Riemann sphere (the Extended Complex Plane), we obtain relations between single qubit unitary operations and Möbius transformations on the Extended Complex Plane.
Bappaditya Bhowmik - One of the best experts on this subject based on the ideXlab platform.
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Loewner chain and quasiconformal extension of some classes of univalent functions
Complex Variables and Elliptic Equations, 2019Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the Extended Complex Plane. Some of these extens...
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On some results for a class of meromorphic functions having quasiconformal extension
Proceedings - Mathematical Sciences, 2018Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:We consider the class \(\Sigma (p)\) of univalent meromorphic functions f on \({\mathbb D}\) having a simple pole at \(z=p\in [0,1)\) with residue 1. Let \(\Sigma _k(p)\) be the class of functions in \(\Sigma (p)\) which have k-quasiconformal extension to the Extended Complex Plane \({\hat{\mathbb C}}\), where \(0\le k < 1\). We first give a representation formula for functions in this class and using this formula, we derive an asymptotic estimate of the Laurent coefficients for the functions in the class \(\Sigma _k(p)\). Thereafter, we give a sufficient condition for functions in \(\Sigma (p)\) to belong to the class \(\Sigma _k(p).\) Finally, we obtain a sharp distortion result for functions in \(\Sigma (p)\) and as a consequence, we obtain a distortion estimate for functions in \(\Sigma _k(p).\)
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Loewner chain and quasiconformal extension of some classes of univalent functions
arXiv: Complex Variables, 2018Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the Extended Complex Plane. Some of these extensions have been obtained by constructing suitable Loewner chains and others have been obtained by applying a well-known result.
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Area distortion under meromorphic mappings with nonzero pole having quasiconformal extension
2017Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:Let $\Sigma_k(p)$ be the class of univalent meromorphic functions defined on $\mathbb{D}$ with $k$-quasiconformal extension to the Extended Complex Plane $\widehat{\mathbb{C}}$, where $0\leq k < 1$. Let $\Sigma_k^0(p)$ be the class of functions $f \in \Sigma_k(p)$ having expansion of the form $f(z)= 1/(z-p) + \sum_{n=1}^{\infty}b_n z^{n}$ on $\mathbb{D}$. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in $\Sigma_k^0(p)$. As a consequence of the obtained results, we present a sharp estimate for the bounds of the Hilbert transform.
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On some results for meromorphic univalent functions having quasiconformal extension
arXiv: Complex Variables, 2017Co-Authors: Bappaditya Bhowmik, Goutam SatpatiAbstract:We consider the class $\Sigma(p)$ of univalent meromorphic functions $f$ on $\ID$ having simple pole at $z=p\in[0,1)$ with residue 1. Let $\Sigma_k(p)$ be the class of functions in $\Sigma(p)$ which have $k$-quasiconformal extension to the Extended Complex Plane $\sphere$ %with $q=\frac{1+k}{1-k}$ where $0\leq k < 1$. We first give a representation formula for functions in this class and using this formula we derive an asymptotic estimate of the Laurent coefficients for the functions in the class $\Sigma_k(p)$. Thereafter we give a sufficient condition for functions in $\Sigma(p)$ to belong in the class $\Sigma_k(p).$ Finally we obtain a sharp distortion result for functions in $\Sigma(p)$ and as a consequence, we get a distortion estimate for functions in $\Sigma_k(p).$
Vladimir V. Sergeichuk - One of the best experts on this subject based on the ideXlab platform.
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Topological classification of Möbius transformations
Journal of Mathematical Sciences, 2013Co-Authors: Tetiana Rybalkina, Vladimir V. SergeichukAbstract:Linear fractional transformations on the Extended Complex Plane are classified up to topological conjugacy. Recall that two transformations f and g are called topologically conjugate if there exists a homeomorphism h such that g = h ^ − 1 ◦ f ◦ h , where ◦ is the composition of mappings.
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Topological classification of Mobius transformations
arXiv: Dynamical Systems, 2013Co-Authors: Tetiana Rybalkina, Vladimir V. SergeichukAbstract:Linear fractional transformations on the Extended Complex Plane are classified up to topological conjugacy. Recall that two transformations f and g are called topologically conjugate if there exists a homeomorphism h such that hg=fh.
