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This paper's main objective is to find new upper bounds for the norm of the sum of two Hilbert space operators and their Kronecker product. The obtained results extend some previously known results from the set of positive operators to arbitrary ones and refine several existing bounds. In particular, applications of the established bounds include refining celebrated numerical radius inequalities and the celebrated operator Cauchy–Schwarz norm inequality. [ABSTRACT FROM AUTHOR]

We present general refinements of the Cauchy–Schwarz and Buzano inequalities over complete inner product spaces showing these improvements have interesting statistical applications. [ABSTRACT FROM AUTHOR]

This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.

We prove a set of inequalities that interpolate the Cauchy-Schwarz inequality and the triangle inequality. Every nondecreasing, convex function with a concave derivative induces such an inequality. They hold in any metric space that satisfies a metric version of the Cauchy-Schwarz inequality, including all CAT (0) spaces and, in particular, all Euclidean spaces. Because these inequalities establish relations between the six distances of four points, we call them quadruple inequalities. In this context, we introduce the quadruple constant a real number that quantifies the distortion of the Cauchy-Schwarz inequality by a given function. Additionally, for inner product spaces, we prove an alternative, more symmetric version of the quadruple inequalities, which generalizes the parallelogram law. [ABSTRACT FROM AUTHOR]

In this paper, we present some weighted Hölder-type inequalities for N −tuple of n × n complex matrices, with applications that include refined bounds for the norm of the sum of two matrices, the numerical radius of a matrix and the numerical radius for the product of two matrices. The relation with the existing literature will be discussed, where some results are proven to be improvements of celebrated results in the literature. Further, numerical examples are given to support the significance of other results. [ABSTRACT FROM AUTHOR]

The main preoccupation of this article is the characterization of Richard's inequality, which it is closely related to Buzano's inequality. Finally, we present a new approach for Richard's inequality, where we use the Selberg operator. [ABSTRACT FROM AUTHOR]

The problem of establishing an upper bound for the volume of a parallelepiped is considered by utilizing an original approach involving a skew-symmetric matrix of order four (along with its Moore–Penrose inverse). It is shown that the commonly known inequality characterizing the bound can be virtually sharpened. Similarly, a sharpening is established with respect to the Cauchy–Schwarz inequality. General properties of the Moore–Penrose inverse of a skew-symmetric matrix are discussed as well.

The nonclassical behavior of the optical model has important applications in numerous quantum information fields. The suggested optical model in this study is based on a five-level atom (5LA) influenced by a two-mode quantized field (2MQF) inside a high-Q cavity that is filled with three types of nonlinear Kerr media. The nonlinear optical model has been solved analytically in which the components of this optical model are in perfect resonance. By considering the initial state of 5LA is in the upper state (excited state), whilst the initial state of 2MQF is in the coherent state. Some useful statistical measurements such as population inversion, Shannon entropy, second-order correlation, as well as Cauchy–Schwarz inequality that corresponds to P-functions are studied and discussed. The effects of the Kerr parameters on these statistical measurements are analyzed. All proposed statistical measurements confirm the existence of nonclassical behavior with time except for some small-time intervals.

The nonclassical behavior of the optical model has important applications in numerous quantum information fields. The suggested optical model in this study is based on a five-level atom (5LA) influenced by a two-mode quantized field (2MQF) inside a high-Q cavity that is filled with three types of nonlinear Kerr media. The nonlinear optical model has been solved analytically in which the components of this optical model are in perfect resonance. By considering the initial state of 5LA is in the upper state (excited state), whilst the initial state of 2MQF is in the coherent state. Some useful statistical measurements such as population inversion, Shannon entropy, second-order correlation, as well as Cauchy–Schwarz inequality that corresponds to P-functions are studied and discussed. The effects of the Kerr parameters on these statistical measurements are analyzed. All proposed statistical measurements confirm the existence of nonclassical behavior with time except for some small-time intervals. [ABSTRACT FROM AUTHOR]

In this work, a refinement of the Cauchy–Schwarz inequality in inner product space is proved. A more general refinement of the Kato's inequality or the so called mixed Schwarz inequality is established. Refinements of some famous numerical radius inequalities are also pointed out. As shown in this work, these refinements generalize and refine some recent and old results obtained in literature. Among others, it is proved that if T ∈ B H , then ω 2 T ≤ 1 12 T + T ∗ 2 + 1 3 ω T T + T ∗ ≤ 1 6 T 2 + T ∗ 2 + 1 3 ω T T + T ∗ , which refines the recent inequality obtained by Kittaneh and Moradi in [10]. [ABSTRACT FROM AUTHOR]

This work presents applications of certain functions to Berezin radius inequalities, as well as further proofs of inequalities linking them. We also establish new inequalities using the operator on a functional Hilbert space concerning the Berezin radius and Berezin norm of specific finite sums. There are also several Berezin radius inequalities provided. [ABSTRACT FROM AUTHOR]

We show that Cauchy-Schwarz inequality for shifted quantum integral operator does not hold in general and we prove the conditions under which it is valid. We apply it to Ostrowski type inequalities for shifted quantum integral operator. [ABSTRACT FROM AUTHOR]

Unit distributions, exhibiting inherent symmetrical properties, have been extensively studied across various fields. A significant challenge in these studies, particularly evident in parameter estimations, is the existence and uniqueness of estimators. Often, it is challenging to demonstrate the existence of a unique estimator. The major issue with maximum likelihood and other estimator-finding methods that use iterative methods is that they need an initial value to reach the solution. This dependency on initial values can lead to local extremes that fail to represent the global extremities, highlighting a lack of symmetry in solution robustness. This study applies a very simple, and unique, estimation method for unit Weibull and unit Burr XII distributions that both attain the global maximum value. Therefore, we can conclude that the findings from the obtained propositions demonstrate that both the maximum likelihood and graphical methods are symmetrically similar. In addition, three real-world data applications are made to show that the method works efficiently. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {L}(\mathscr {H}) $ L (H) denote the $ C^* $ C ∗ -algebra of adjointable operators on a Hilbert $ C^* $ C ∗ -module $ \mathscr {H} $ H . In this paper, we introduce the generalized Cauchy–Schwarz inequality for operators in $ \mathscr {L}(\mathscr {H}) $ L (H). More precisely, an operator $ A\in \mathscr {L}(\mathscr {H}) $ A ∈ L (H) is said to satisfy the generalized Cauchy–Schwarz inequality if there exists $ \nu \in (0, 1) $ ν ∈ (0 , 1) such that \[ \|\langle Ax, y\rangle\|\leq (\|Ax\|\|y\|)^{\nu}(\|Ay\|\|x\|)^{1 - \nu} \quad (x, y \in \mathscr{H}). \] ‖ ⟨ A x , y ⟩ ‖ ≤ (‖ A x ‖ ‖ y ‖) ν (‖ A y ‖ ‖ x ‖) 1 − ν (x , y ∈ H). We investigate various properties of operators which satisfy the generalized Cauchy–Schwarz inequality. In particular, we prove that if A satisfies the generalized Cauchy–Schwarz inequality such that A has the polar decomposition, then A is paranormal. In addition, we show that if for A the equality holds in the generalized Cauchy–Schwarz inequality, then A is cohyponormal. Among other things, when A has the polar decomposition, we prove that A is semi-hyponormal if and only if $ \left \|\langle Ax, y\rangle \right \| \leq \left \|{|A|}^{1/2}x\right \|\left \|{|A|}^{1/2}y\right \| $ ‖ ⟨ A x , y ⟩ ‖ ≤ ‖ | A | 1 / 2 x ‖ ‖ | A | 1 / 2 y ‖ for all $ x, y \in \mathscr {H} $ x , y ∈ H . [ABSTRACT FROM AUTHOR]

In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator Hölder type inequalities. Among other results, it is shown that if A = (At)t∈Ω and B=(Bt)t∈Ω are continuous fields of positive invertible operators in a unital C*-algebra A such that ∫Ω At dμ(t) ≤ IA and ∫Ω Bt dμ(t) ≤ IA, and if ωf is an arbitrary operator mean with the representing function f, then ...for all 0 < p ≤ 1, which is an extension of the operator Bellman inequality. [ABSTRACT FROM AUTHOR]

We have analysed the Cauchy–Schwarz inequality for the α -th moment of the speed of the molecules of an ideal gas in d-dimension following the Maxwell distribution of speeds and also for the molecules effusing through the hole. In the later case the distribution is non-Maxwellian. We have also studied the energy fluctuations for both cases in d-dimension and examined the role of the spatial dimensionality of the system on the distribution. [ABSTRACT FROM AUTHOR]

New extensions of the Cauchy–Schwarz inequality in the framework of pre-Hilbert C ∗ -modules are given. An application to the numerical radius in C ∗ -algebras is also provided. [ABSTRACT FROM AUTHOR]

This paper introduces several refinements of the classical Selberg inequality, which is considered a significant result in the study of the spectral theory of symmetric spaces, a central topic in the field of symmetry studies. By utilizing the contraction property of the Selberg operator, we derive improved versions of the classical Selberg inequality. Additionally, we demonstrate the interdependence among well-known inequalities such as Cauchy–Schwarz, Bessel, and the Selberg inequality, revealing that these inequalities can be deduced from one another. This study showcases the enhancements made to the classical Selberg inequality and establishes the interconnectedness of various mathematical inequalities. [ABSTRACT FROM AUTHOR]

The purpose of this research paper is to introduce new Cauchy–Schwarz inequalities that are valid in semi-Hilbert spaces, which are generalizations of Hilbert spaces. We demonstrate how these new inequalities can be employed to derive novel A-numerical radius inequalities, where A denotes a positive semidefinite operator in a complex Hilbert space. Some of our novel A-numerical radius inequalities expand upon the existing literature on numerical radius inequalities with Hilbert space operators, which are important tools in functional analysis. We use techniques from semi-Hilbert space theory to prove our results and highlight some applications of our findings. [ABSTRACT FROM AUTHOR]

We present a generic refinement to the Cauchy-Schwarz inequality in both inner product space and probability space and study some of its special cases. [ABSTRACT FROM AUTHOR]

The main goal of this work is to present new matrix inequalities of Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that ||T||≤1/4(|| |T|+|T*|+2RT||+|| T|+|T*|-2RT||) where RT is the real part of the matrix T. [ABSTRACT FROM AUTHOR]

We introduce a generic inequality of the form f 2 (x) ≤ f (s) f (2 x − s) and show that many special functions such as gamma and polygamma functions, Riemann zeta function, beta and incomplete beta functions, Gauss and confluent hypergeometric functions, elliptic integrals and error function satisfy such type of the inequality. We also obtain new inequalities of this type for inverse trigonometric functions arcsin ⁡ x and arctan ⁡ x. [ABSTRACT FROM AUTHOR]

In this note, we introduce an idea of deriving some inequalities, using the discriminants of polynomials. This gives another proof of the famous Hölder's inequality. [ABSTRACT FROM AUTHOR]

In this paper, we study the norm and skew angular distances in a normed space X , where convex functions are used to obtain refinements and reverses of some outstanding results in the literature. For example, in this regard, we show that if a , b ∈ X are non-zero and if p , q > 0 are such that 1 p + 1 q = 1 , then 2 λ p r a r + q r b r 2 - p a + q b 2 r ≤ p r - 1 a r + q r - 1 b r - a + b r ≤ 2 μ p r a r + q r b r 2 - p a + q b 2 r , where r ≥ 1 , λ = min 1 / p , 1 / q and μ = max 1 / p , 1 / q . Then we explain how this result extends some known results in the literature. Many other related results will be also shown. Then, with the theme of convexity, we employ a log-convex approach on certain matrix functions to obtain improvements and new sights of some matrix inequalities, including possible bounds of ‖ A t X B 1 - t ‖ , where A, B are positive definite matrices, X is an arbitrary matrix, ‖ · ‖ is a unitarily invariant norm and 0 ≤ t ≤ 1. Many other results involving matrix and scalar log-convex functions will be presented too. [ABSTRACT FROM AUTHOR]

The Berezin transform ˜A and the Berezin number of an operator A on the reproducing kernel Hilbert space over some set Ω with normalized reproducing kernel k̂λ are defined, respectively, by ˜A(λ) = ⟨ Ak̂λ, k̂λ ⟩, λ ∈ Ω and ber ( A ) := supλ∈Ω ∣ ˜A(λ) ∣. A straightforward comparison between these characteristics yields the inequalities ber (A) ≤ 1/2 ( ∥ A ∥ber + ∥ A 2 ∥ber1/2) . In this paper, we study further inequalities relating them. Namely, we obtained some refinement of Berezin number inequalities involving convex functions. In particular, for A ∈ B (H) and r ≥ 1 we show that ber2r (A) ≤ 1/4 ( ∥ A* A + A A* ∥berr + ∥ A* A − A A* ∥berr ) + 1/2 berr ( A² ) . [ABSTRACT FROM AUTHOR]

We give a simple proof of the existence of a complex eigenvalue of real or complex square matrices with Linear Algebra accessible to students beginning a first course on the subject, namely using just the Cauchy–Schwarz inequality in C n , linear independence and the matrix product, together with the Weierstrass extreme value theorem. It yields an equally simple proof of the Fundamental Theorem of Algebra with Linear Algebra. [ABSTRACT FROM AUTHOR]

A scheme is proposed to realize the non‐classical correlation between magnons and photons in a cavity optomagnonic system, which supports both photon modes and a magnon mode. Starting with the system being initially prepared in its ground state, two laser pulses successively drive corresponding optical mode. A two‐mode squeezing interaction between optical mode 1 and the magnon mode is created by the first pulse, which leads to a non‐classical correlation between photons and magnons. To verify this non‐classical correlation, the second pulse is utilized to transfer the magnon state to another optical mode, thus the correlated photon–photon pairs are generated out of the cavity. By discussing the violation of Cauchy–Schwarz inequality, based on numerical simulation, it is confirmed that non‐classical correlated photon–magnon pairs can be created in the weak coupling regime, which relaxes the requirements of experimental conditions. The result indicates that cavity optomagnonics can be a promising platform for studying magnon‐based quantum information processing. [ABSTRACT FROM AUTHOR]

In this paper, we present some generalizations and improvements of a new integral inequality from the 29th IMC in 2022. Some applications of our new results are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we prove a type of Selberg type inequality in a 2 -*-semi inner product A-module over a C*-algebra A. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method.

In this paper, we give an explicit upper bound on h (p) , the least primitive root modulo p 2 . Since a primitive root modulo p is not primitive modulo p 2 if and only if it belongs to the set of integers less than p which are p th power residues modulo p 2 , we seek the bounds for N 1 (H) and N 2 (H) to find H which satisfies N 1 (H) − N 2 (H) > 0 , where, N 1 (H) denotes the number of primitive roots modulo p not exceeding H , and N 2 (H) denotes the number of p th powers modulo p 2 not exceeding H. The method we mainly use is to estimate the character sums contained in the expressions of the N 1 (H) and N 2 (H) above. Finally, we show that h (p) < p 0. 7 4 for all primes p. This improves the recent result of Kerr et al. [ABSTRACT FROM AUTHOR]

The Cauchy-Schwarz, Buzano and Kreĭn inequalities are three inequalities about inner product. The main goal of this article is to present refinements of Buzano and Cauchy-Schwarz inequalities, and to present a new proof of a refined version of a Kreĭn-type inequality. Applications that include Buzano-type inequalities for certain operators, operator norm and numerical radius inequalities of Hilbert space operators will be presented. [ABSTRACT FROM AUTHOR]

In this paper, we will be concerned with the average behavior of the n th normalized Fourier coefficients of symmetric square L-function (i.e., L (s , s y m 2 f) ) over certain sequence of positive integers. Precisely, we prove an asymptotic formula for ∑ a 2 + b 2 + c 2 + d 2 ≤ x (a , b , c , d) ∈ Z 4 λ s y m 2 f 2 (a 2 + b 2 + c 2 + d 2) , where x ≥ x 0 (sufficiently large), and L (s , s y m 2 f) : = ∑ n = 1 ∞ λ s y m 2 f (n) n s . [ABSTRACT FROM AUTHOR]

We derive strong variance-based uncertainty relations for arbitrary two and more unitary operators by re-examining the mathematical foundation of the uncertainty relation. This is achieved by strengthening the celebrated Cauchy–Schwarz inequality using a method of brackets and convex functions. The unitary uncertainty relations outperform several strong unitary uncertainty relations, notably better than some recent best lower bounds such as [Phys. Rev. Lett. 120, 230402 (2018)] and [Phys. Rev. A. 100, 022116 (2019)]. [ABSTRACT FROM AUTHOR]

In the paper, we introduce a new partial derivative call it β-partial derivatives as the most natural extensions of the limit definitions of the partial derivative and the β-derivative, which obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple type inequalities for the β-partial derivatives.

Connectivity is a critical parameter which can measure the reliability of networks. Let T be a vertex set of G. If G − T has at least t components, then T is a t -component cut of G. The t -component connectivity κ t (G) of G is the vertex number of a smallest t -component cut. Cartesian product of graphs is a useful method to construct a large network. We will use Cauchy–Schwarz inequality to determine the component connectivity of Cartesian product of some graphs. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of scalability to higher dimensional space in blind source separation (BSS), where the number of sources is greater than two. Herein, we propose two schemes of pairwise non-parametric independent component analysis (ICA) algorithms based on Convex Cauchy–Schwarz Divergence (CCS–DIV) for high dimensional problem in BSS. We extend the pairwise method to the scenario of more than two sources, two improved ICA algorithms are developed. Moreover, we employ adaptive sampling technique that samples the signal into small time blocks to evaluate the integration of the CCS–DIV and reduce the computational complexity. The two presented methods enable fast and efficient demixing of sources in real-world high dimensional source applications. Finally, we demonstrate with simulations including a wide variety of source distributions, showing that our presented methods outperform many of the presently known methods in terms of performance and computational complexity. [ABSTRACT FROM AUTHOR]

In this note, we study Cauchy-Schwarz-type inequality for fractional Strum-Liouville boundary value problem containing Caputo derivative of order α, 1 < α ≤ 2. A lower bound for the smallest eigenvalue is determined using this inequality. We give a comparison between the smallest eigenvalue and its lower bound obtained from the Lyapunov-type and Cauchy-Schwarz-type inequalities which indicate the properties of eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, a series of new inequalities for definite integral are constructed. By rigorous analysis, the sufficient and necessary conditions for which the equality holds are carried out. Additionally, the optimal constant factors for these new inequalities are obtained. [ABSTRACT FROM AUTHOR]

Making use of the q - difference operator L p , q a , c , we introduce a new two subclasses of p - valent analytic functions in the open unit disk. The main objective of the present paper is to investigate the various important properties and characteristics of each of these subclasses. Furthermore, several properties involving neighborhoods and modified Hadamard products of functions in these subclasses are obtained. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present some new Cauchy–Schwarz inequalities for N-tuple diamond-alpha integral on time scales. The obtained results improve and generalize some Cauchy–Schwarz type inequalities given by many authors.

Graph theory offers experts with a useful tool called the topological index which analyses the characteristics of chemical compounds. Motivated by the recent work of redefined Zagreb indices ( R e Z G 1 , R e Z G 2 , R e Z G 3 ), novel topological indices such as Neighbourhood redefined first and second Zagreb indices ( N R e Z 1 , N R e Z 2 ) are proposed and computed for a molecular graph G of two carborundum structures. The correlation coefficient of N R e Z 1 , N R e Z 2 with ten properties of octane isomers are determined, of which entropy and acentric factor showed a good correlation. Also, the lower and upper bounds of redefined Zagreb indices are determined using classic results, Pólya–Szegö and Cauchy–Schwarz inequalities for simple graphs and trees. [ABSTRACT FROM AUTHOR]

In this short communication we remark that the well-known Cauchy–Schwarz inequality for expectations of random variables is a consequence of Jensen's inequality, which does not seem to have appeared previously in the literature. [ABSTRACT FROM AUTHOR]

We study some interesting characterizations of real inner product spaces expressed in terms of angular distances. We first discuss the equivalence of characterizing an inner product space via the usual angular distance and the p-angular distance. Then, we establish a parametric family of upper bounds for the usual angular distance which also serves as a characterization of an inner product space. As an application, bounds for the usual angular distance are utilized in obtaining improvements of the real Cauchy–Schwarz inequality. Finally, we give several comparative relations for angular distances in inner product spaces. [ABSTRACT FROM AUTHOR]

In this article, we study a generalization of the n-inner product which we name weak n-inner product. As particular case, we consider the n-iterated 2-inner product and we give its representation in terms of the standard k-inner products, k ≤ n , using the Dodgson’s identity for determinants. Finally, we present several applications, including a brief characterization of a linear regression model for the random variables in discrete case and a generalization of the Chebyshev functional using the n-iterated 2-inner product. [ABSTRACT FROM AUTHOR]

In this paper, a stabilized numerical method with high accuracy is proposed to solve time-fractional singularly perturbed convection-diffusion equation with variable coefficients. The tailored finite point method (TFPM) is adopted to discrete equation in the spatial direction, while the time direction is discreted by the G-L approximation and the L1 approximation. It can effectively eliminate non-physical oscillation or excessive numerical dispersion caused by convection dominant. The stability of the scheme is verified by theoretical analysis. Finally, one-dimensional and two-dimensional numerical examples are presented to verify the efficiency of the method. [ABSTRACT FROM AUTHOR]

We consider the equality case in Thunsdorff inequality and Cauchy-Schwarz inequality. For these two equations we prove Ulam stability. [ABSTRACT FROM AUTHOR]

Externally induced entanglement amplification in a coherently pumped correlated emission laser (CEL) with parametric amplifier and coupled to a two-mode squeezed vacuum reservoir is presented. The combination of the master equation and stochastic differential equation is employed to investigate the entanglement of the two-mode light generated by the quantum system. The resulting solutions of the correlation properties of noise forces associated with the normal ordering are used to find the mean photon number of the cross-correlated mode and separate cavity modes, quadrature fluctuations, smallest eigenvalue of the symplectic matrix, and photon number correlation function. It is found that pumping atoms from the lower energy state to excited state results in a robust entanglement that remains in its maximum strength over a wide range of the strong classical driving radiation. The introduction of the nonlinear crystal into the linear cavity, and coupling the system to the two-mode squeezed vacuum environment lead to a significant enhancement of entanglement of the cavity light. The enhanced entanglement is quantified employing logarithmic negativity, Cauchy–Schwarz inequality, and Duan et al. criteria. It has been observed that an intense light can be produced where the entanglement is strong. [ABSTRACT FROM AUTHOR]