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Search Results (7)

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Keywords = de Bruijn inequality

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22 pages, 312 KiB  
Article
Selberg’s Inequality and Selberg Operator Bounds in Hilbert Spaces with Applications
by Salma Aljawi, Cristian Conde, Silvestru Sever Dragomir and Kais Feki
Axioms 2025, 14(8), 575; https://doi.org/10.3390/axioms14080575 - 25 Jul 2025
Viewed by 225
Abstract
In the present work, we give a new proof of the well-known Selberg’s inequality in complex Hilbert spaces from an operator-theoretic perspective, establishing its fundamental equivalence with the Cauchy–Bunyakovsky–Schwarz inequality. We also derive several lower and upper bounds for the Selberg operator, including [...] Read more.
In the present work, we give a new proof of the well-known Selberg’s inequality in complex Hilbert spaces from an operator-theoretic perspective, establishing its fundamental equivalence with the Cauchy–Bunyakovsky–Schwarz inequality. We also derive several lower and upper bounds for the Selberg operator, including its norm estimates, refining classical results such as de Bruijn’s and Bohr’s inequalities. Additionally, we revisit a recent claim in the literature, providing a clarification of the conditions under which Selberg’s inequality extends to abstract bilinear forms. Full article
(This article belongs to the Section Mathematical Analysis)
12 pages, 259 KiB  
Article
Some Refinements and Generalizations of Bohr’s Inequality
by Salma Aljawi, Cristian Conde and Kais Feki
Axioms 2024, 13(7), 436; https://doi.org/10.3390/axioms13070436 - 28 Jun 2024
Cited by 1 | Viewed by 770
Abstract
In this article, we delve into the classic Bohr inequality for complex numbers, a fundamental result in complex analysis with broad mathematical applications. We offer refinements and generalizations of Bohr’s inequality, expanding on the established inequalities of N. G. de Bruijn and Radon, [...] Read more.
In this article, we delve into the classic Bohr inequality for complex numbers, a fundamental result in complex analysis with broad mathematical applications. We offer refinements and generalizations of Bohr’s inequality, expanding on the established inequalities of N. G. de Bruijn and Radon, as well as leveraging the class of functions defined by the Daykin–Eliezer–Carlitz inequality. Our novel contribution lies in demonstrating that Bohr’s and Bergström’s inequalities can be derived from one another, revealing a deeper interconnectedness between these results. Furthermore, we present several new generalizations of Bohr’s inequality, along with other notable inequalities from the literature, and discuss their various implications. By providing more comprehensive and verifiable conditions, our work extends previous research and enhances the understanding and applicability of Bohr’s inequality in mathematical studies. Full article
(This article belongs to the Special Issue Current Research on Mathematical Inequalities II)
16 pages, 279 KiB  
Article
On Variance and Average Moduli of Zeros and Critical Points of Polynomials
by Sajad A. Sheikh, Mohammad Ibrahim Mir, Osama Abdulaziz Alamri and Javid Gani Dar
Symmetry 2024, 16(3), 349; https://doi.org/10.3390/sym16030349 - 14 Mar 2024
Cited by 1 | Viewed by 1364
Abstract
This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we [...] Read more.
This paper investigates various aspects of the distribution of roots and critical points of a complex polynomial, including their variance and the relationships between their moduli using an inequality due to de Bruijn. Making use of two other inequalities-again due to de Bruijn-we derive two probabilistic results concerning upper bounds for the average moduli of the imaginary parts of zeros and those of critical points, assuming uniform distribution of the zeros over a unit disc and employing the Markov inequality. The paper also provides an explicit formula for the variance of the roots of a complex polynomial for the case when all the zeros are real. In addition, for polynomials with uniform distribution of roots over the unit disc, the expected variance of the zeros is computed. Furthermore, a bound on the variance of the critical points in terms of the variance of the zeros of a general polynomial is derived, whereby it is established that the variance of the critical points of a polynomial cannot exceed the variance of its roots. Finally, we conjecture a relation between the real parts of the zeros and the critical points of a polynomial. Full article
(This article belongs to the Section Mathematics)
9 pages, 270 KiB  
Article
Skewed Jensen—Fisher Divergence and Its Bounds
by Takuya Yamano
Foundations 2021, 1(2), 256-264; https://doi.org/10.3390/foundations1020018 - 16 Nov 2021
Cited by 3 | Viewed by 3465
Abstract
A non-uniform (skewed) mixture of probability density functions occurs in various disciplines. One needs a measure of similarity to the respective constituents and its bounds. We introduce a skewed Jensen–Fisher divergence based on relative Fisher information, and provide some bounds in terms of [...] Read more.
A non-uniform (skewed) mixture of probability density functions occurs in various disciplines. One needs a measure of similarity to the respective constituents and its bounds. We introduce a skewed Jensen–Fisher divergence based on relative Fisher information, and provide some bounds in terms of the skewed Jensen–Shannon divergence and of the variational distance. The defined measure coincides with the definition from the skewed Jensen–Shannon divergence via the de Bruijn identity. Our results follow from applying the logarithmic Sobolev inequality and Poincaré inequality. Full article
(This article belongs to the Section Information Sciences)
32 pages, 753 KiB  
Article
ϕ-Informational Measures: Some Results and Interrelations
by Steeve Zozor and Jean-François Bercher
Entropy 2021, 23(7), 911; https://doi.org/10.3390/e23070911 - 18 Jul 2021
Cited by 3 | Viewed by 3403
Abstract
In this paper, we focus on extended informational measures based on a convex function ϕ: entropies, extended Fisher information, and generalized moments. Both the generalization of the Fisher information and the moments rely on the definition of an escort distribution linked to [...] Read more.
In this paper, we focus on extended informational measures based on a convex function ϕ: entropies, extended Fisher information, and generalized moments. Both the generalization of the Fisher information and the moments rely on the definition of an escort distribution linked to the (entropic) functional ϕ. We revisit the usual maximum entropy principle—more precisely its inverse problem, starting from the distribution and constraints, which leads to the introduction of state-dependent ϕ-entropies. Then, we examine interrelations between the extended informational measures and generalize relationships such the Cramér–Rao inequality and the de Bruijn identity in this broader context. In this particular framework, the maximum entropy distributions play a central role. Of course, all the results derived in the paper include the usual ones as special cases. Full article
(This article belongs to the Special Issue Entropies, Divergences, Information, Identities and Inequalities)
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24 pages, 625 KiB  
Article
From Rényi Entropy Power to Information Scan of Quantum States
by Petr Jizba, Jacob Dunningham and Martin Prokš
Entropy 2021, 23(3), 334; https://doi.org/10.3390/e23030334 - 12 Mar 2021
Cited by 5 | Viewed by 4650
Abstract
In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also [...] Read more.
In this paper, we generalize the notion of Shannon’s entropy power to the Rényi-entropy setting. With this, we propose generalizations of the de Bruijn identity, isoperimetric inequality, or Stam inequality. This framework not only allows for finding new estimation inequalities, but it also provides a convenient technical framework for the derivation of a one-parameter family of Rényi-entropy-power-based quantum-mechanical uncertainty relations. To illustrate the usefulness of the Rényi entropy power obtained, we show how the information probability distribution associated with a quantum state can be reconstructed in a process that is akin to quantum-state tomography. We illustrate the inner workings of this with the so-called “cat states”, which are of fundamental interest and practical use in schemes such as quantum metrology. Salient issues, including the extension of the notion of entropy power to Tsallis entropy and ensuing implications in estimation theory, are also briefly discussed. Full article
(This article belongs to the Special Issue The Statistical Foundations of Entropy)
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9 pages, 91 KiB  
Article
An Integral Representation of the Relative Entropy
by Miku Hirata, Aya Nemoto and Hiroaki Yoshida
Entropy 2012, 14(8), 1469-1477; https://doi.org/10.3390/e14081469 - 8 Aug 2012
Cited by 5 | Viewed by 5407
Abstract
Recently the identity of de Bruijn type between the relative entropy and the relative Fisher information with the reference moving has been unveiled by Verdú via MMSE in estimation theory. In this paper, we shall give another proof of this identity in more [...] Read more.
Recently the identity of de Bruijn type between the relative entropy and the relative Fisher information with the reference moving has been unveiled by Verdú via MMSE in estimation theory. In this paper, we shall give another proof of this identity in more direct way that the derivative is calculated by applying integrations by part with the heat equation. We shall also derive an integral representation of the relative entropy, as one of the applications of which the logarithmic Sobolev inequality for centered Gaussian measures will be given. Full article
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