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We establish necessary and sufficient conditions for the one-dimensional differential Hardy inequality to hold, including the overdetermined case. The solution is given in terms different from those of the known results. Moreover, the least constant for this inequality is estimated.

In this paper, we will prove some new dynamic inequalities of Hilbert's type on time scales. Our results as special cases extend some obtained dynamic inequalities on time scales.and also contain some integral and discrete in- equalities as special cases. We prove our main results by using some algebraic inequalities, H?older's inequality, Jensen's inequality and a simple consequence of Keller's chain rule on time scales.

The well-known arithmetic-geometric mean inequality for singular values, due to Bhatia and Kittaneh, is one of the most important singular value inequalities for compact operators. The purpose of this study is to give new singular value inequalities for compact operators and prove that these inequalities are equivalent to arithmetic-geometric mean inequality, the way by which several future studies could be done.

We refine some classical inequalities for trigonometric functions, such as Jordan's inequality, Cusa–Huygens's inequality, and Kober's inequality.

A new finite-sum inequality is derived that includes the discrete Jensen's inequality and the Abel lemma-based finite-sum inequality as special cases. Another new inequality, which needs fewer decision variables than the first one and provides a tighter lower bound than the Abel lemma-based finite-sum inequality, is also given. Applying these new inequalities yields new results on stability analysis for discrete time-delay systems. Two numerical examples demonstrate the effectiveness and superiority of the proposed methods.

In this paper we establish the best constant of an anisotropic Gagliardo–Nirenberg-type inequality related to the Benjamin–Ono–Zakharov–Kuznetsov equation. As an application of our results, we prove the uniform bound of solutions for such a equation in the energy space.

In this paper, by using Jensen’s inequality and Chebyshev integral inequality, some generalizations and new refined Hardy type integral inequalities are obtained. In addition, the corresponding reverse relation are also proved.

In this paper, based on the discrete Wirtinger inequality, a novel summation inequality is established which extends the Jensen inequality. By the technique of the novel inequality, a sufficient criterion on asymptotic stability of discrete neural networks with time-varying delays is obtained. A numerical example is given to demonstrate the effectiveness of the inequality.

We study a certain improved fractional Sobolev–Poincaré inequality on domains, which can be considered as a fractional counterpart of the classical Sobolev–Poincaré inequality. We prove the equivalence of the corresponding weak and strong type inequalities; this leads to a simple proof of a strong type inequality on John domains. We also give necessary conditions for the validity of an improved fractional Sobolev–Poincaré inequality, in particular, we show that a domain of finite measure, satisfying this inequality and a ‘separation property’, is a John domain.

In this paper, we focus on a decomposition property recently introduced in the inequality literature and known as the weak decomposition. Such a property provides interesting analyses by allowing one to separate the within-group contribution to total inequality from the between-group contribution. A limitation of the current method of decomposition is that, depending on the structure–absolute, relative, compromise–of the inequality index, specific weights have to be used. To avoid such a problem, we propose a unique decomposition property where the weighting functions depend on the size of the population and the mean income. This allows us to characterize a large family of weakly decomposable inequality indices without any recourse to implicit invariance value judgments.

Audenaert recently obtained an inequality for unitarily invariant norms that interpolates between the arithmetic–geometric mean inequality and the Cauchy–Schwarz inequality for matrices. A refined version of Audenaert’s inequality for the Hilbert–Schmidt norm is given. Other interpolating inequalities for unitarily invariant norms are also presented.

In this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space H we have the inequality 1 2 [ 〈 | A | 2 + | B | 2 2 x , x 〉 1 / 2 〈 | A | 2 + | B | 2 2 y , y 〉 1 / 2 + | 〈 | A | 2 + | B | 2 2 x , y 〉 | ] ≥ | 〈 Re ( B * A ) x , y 〉 | $${1 \over 2}\left[ {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{x}},{\rm{x}}} \right\rangle ^{1/2} \left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{y}},{\rm{y}}} \right\rangle ^{1/2} + \left| {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over {\rm{2}}}} {\rm{x}},{\rm{y}}\right\rangle } \right|} \right] \ge \left| {\left\langle {{\mathop{\rm Re}\nolimits} ({\rm{B}}*{\rm{A}})\,{\rm{x}},{\rm{y}}} \right\rangle } \right|$$ for A, B two bounded linear operators on H such that Re (B*A) is a nonnegative operator and any vectors x, y ∈ H. Applications for norm and numerical radius inequalities are given as well.

In this article, we present multiple-term refinements of Young's inequality for real numbers and operators. In particular, given a natural number N , we find N positive terms refining Young's inequality. Detailed properties of this refinement are studied leading to a new interesting mean.

We prove a generalization of Jackson's inequality for compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e.?the well known Holder Continuity Property (HCP) and the less known but crucial Łojasiewicz-Siciak inequality (ŁS). Moreover, we show that ŁS is a necessary condition for our Jackson type inequality. The resulting Jackson Property provides us with a large class of sets in the complex plane for which the global and local Markov inequalities are equivalent.

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of Hardy inequality and Moser–Trudinger inequality on M. Furthermore, the constant we obtain is sharp.

A Buzano type inequality for two nonnegative Hermitian forms is obtained. Applications to inequalities for norm and numerical radius of bounded linear operators in complex Hilbert spaces are given.

By applying the weight functions, the technique of real analysis and Hermite-Hadamard’s inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of exponential function with the best possible constant factor expressed by the gamma function is given. The more accurate equivalent forms, the operator expressions with the norm, the reverses, and some particular cases are considered.

In this note, a generalized form of the celebrated Berry-Esseen inequality is developed assuming only second-order moments to exist in the case of independent but not identically distributed random variables. The result generalizes and unifies many well known and highly used forms of the Berry-Esseen inequality.

By the use of weight coefficients and technique of real analysis, a discrete Hilbert-type inequality in the whole plane with multi-parameters and a best possible constant factor is given. The equivalent forms, the operator expressions, and a few particular inequalities are considered.

Based on the free-weighting matrix and integral-inequality methods, a free-matrix-based double integral inequality is proposed in this paper, which includes the Wirtinger-based double integral inequality as a special case. By introducing some free matrices into the inequality, more freedom can be provided in bounding the quadratic double integral. The connection of the new integral inequality and Wirtinger-based double one is well described, which gives a sufficient condition for the application of the new inequality to be less conservative. Furthermore, to investigate the effectiveness of the proposed inequality, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. Numerical examples are given to demonstrate the advantages of the proposed method.

In this article, we derive several multidimensional Hilbert-type inequalities, including certain differential operators. Further, we determine the conditions under which the constants appearing on the right-hand sides of the established inequalities are the best possible. As an application, some particular examples are also studied.

We present a multidimensional integral inequality related to the Hilbert-type inequality and the Carlson-type inequality. As an application, we obtain a sharper form of the Hilbert-type inequality. Carlson-type inequality is also considered.

BACKGROUND Inter - conversion processes of labile molecules obey similar rules to those of reversible chemical reactions. Solving the corresponding linear differential systems is used along this work, as well as in the preceding version. OBJECTIVE The main purpose of the present mini revue paper is to recall, improve and correct some mathematical methods in determining the optimal values at equilibrium, and remarkable particular rate constants. This part was not proved correctly in my previous work. METHOD In my previous work, the proof of the equality of the concentrations of the main species at equilibrium was not correct. In the current manuscript, we use increasing velocity in order to obtain this first important result. To this aim, one applies Schwarz inequality and the case when equality occurs. In order to determine significant rate constants, we characterize these special values in terms of the norm of the linear operator defined by the matrix of the differential system. In my previous work, the normal probability density function was used. The latter method was not realistic. RESULTS Increasing the velocity, one obtains equal optimal values of the concentrations at equilibrium. This method represents a patent in the field. Secondly, characterization of remarkable rate constants (which are also equal) is deduced. The optimal solutions are written explicitly. CONCLUSION Under suitable conditions, the values at equilibrium and the rate constants are equal. The common value at equilibrium equals the common value of the rate constants.

In the present paper, the inequality inverse to the Cauchy–Bunyakovskii–Schwarz inequality and generalizing other well-known inversions of this inequality is proved.

We present a new proof of Carlson–Levin inequality and some of its extensions, based on a dynamic programing-type approach.

In this paper we prove several fractional quantum integral inequalities for the new q-shifting operator ${_{a}}\Phi_{q}(m) = qm + (1-q)a$ introduced in Tariboon et al. (Adv. Differ. Equ. 2015:18, 2015), such as: the q-Holder inequality, the q-Hermite-Hadamard inequality, the q-Cauchy-Bunyakovsky-Schawrz integral inequality, the q-Gruss integral inequality, the q-Gruss-Cebysev integral inequality, and the q-Polya-Szego integral inequality.

By using weight coefficients, technique of real analysis, and Hermite-Hadamard’s inequality, we give a more accurate Hardy-Mulholland-type inequality with multiparameters and a best possible constant factor related to the beta function. The equivalent forms, the reverses, the operator expressions, and some particular cases are also considered.

If $\begin{bmatrix}A & X \\ X^* & B\end{bmatrix}$ is positive semidefinite with each block $n\times n$, we prove that $$2s_j\Big(\Phi(X)\Big)\le s_j\Big(\Phi(A+B)\Big), \qquad j=1, \ldots, n,$$ where $\Phi: X\mapsto X+(\tr X)I$ and $s_j(\cdot)$ means the $j$-th largest singular value. This confirms a conjecture of the author in [Linear Algebra Appl. 459 (2014) 404-410].

The main aim of this survey article is to present recent developments of matrix versions of the arithmetic–geometric mean inequality. Among others, we show improvements and generalizations of the arithmetic–geometric mean inequality for unitarily invariant norms via the Hadamard product, and for singular values via the operator monotone functions.

A novel biased proportional navigation guidance (BPNG) law is proposed for the close approach phase, which aims to make the spacecraft rendezvous with the target in specific relative range and direction. Firstly, in order to describe the special guidance requirements, the concept of zero effort miss vector is proposed and the dangerous area where there exists collision risk for safety consideration is defined. Secondly, the BPNG, which decouples the range control and direction control, is designed in the line-of-sight (LOS) rotation coordinate system. The theoretical analysis proves that BPNG meets guidance requirements quite well. Thirdly, for the consideration of fuel consumption, the optimal biased proportional navigation guidance (OBPNG) law is derived by solving the Schwartz inequality. Finally, simulation results show that BPNG is effective for the close approach with the ability of evading the dangerous area and OBPNG consumes less fuel compared with BPNG.

Positivity of a $2\times2$ operator matrix $[\begin{smallmatrix}A&B\\B^{*}&C\end{smallmatrix}]\geq0$ implies $\sqrt{\|A\|\cdot\|C\|}\geq\|B\|$ for operator norm $\|\cdot\|$ . This can be considered as an operator version of the Schwarz inequality. In this situation, for $A,C\geq0$ , there is a natural notion of geometric mean $A\sharp C$ , for which $\sqrt{\|A\|\cdot\|C\|}\geq\|A\sharp C\|$ . In this paper, we study under what conditions on $A$ , $B$ , and $C$ or on $B$ alone the norm inequality $\sqrt{\|A\|\cdot\|C\|}\geq\|B\|$ can be improved as $\|A\sharp C\|\geq\|B\|$ .

Considering different parameters, Hilbert-type integral inequality for functions f(x), g(x) in L2[0, ∞) will be generalized.

In this paper, a boundary version of the Schwarz inequality is investigated. We obtain more general results at the boundary. If we know the second coefficient in the expansion of the function f(z) = 1 + cpzp + cp + 1zp + 1…, then we obtain new inequalities of the Schwarz inequality at boundary by taking into account cp + 1 and zeros of the function f(z) − 1. The sharpness of these inequalities is also proved.

In this article, we established new results related to a 2-pre-Hilbert space. Among these results we will mention the Cauchy-Schwarz inequality. We show several applications related to some statistical indicators as average, variance and standard deviation and correlation coefficient, using the standard 2-inner product and some of its properties. We also present a brief characterization of a linear regression model for the random variables in discrete case.

The standard proof of the inequality of the title may appear a bit dry, and so I offer one that uses water. Consider n cylindrical cans with different cross-sectional areas ak that are connected by...

Employing a generalized Riccati transformation and integral averaging technique, we show that all solutions of the higher order nonlinear delay differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ y^{(n+2)}(t)+p(t)y^{(n)}(t)+q(t)f\bigl(y\bigl(g(t)\bigr)\bigr)=0 $$\end{document}y(n+2)(t)+p(t)y(n)(t)+q(t)f(y(g(t)))=0 will converge to zero or oscillate, under some conditions listed in the theorems of the present paper. Several examples are also given to illustrate the applications of these results.

We observe a connection between Cauchy–Schwarz’s and Richard’s inequalities in inner product spaces and a Ulam-type stability problem for the multiplicative Sincov functional equation. We prove that this equation is superstable for unbounded mappings, i.e., every unbounded approximate solution is an exact solution.

Let G be a Strongly Regular Graph of order n and A its matrix of adjacency. By the analysis of the spectra of some elements of the Euclidean Jordan Algebra 𝒜 spanned by the powers of A and by the identity In we establish some admissibility conditions on the parameters of the strongly regular graph G using an asymptotic approach. Finally, some admissibility conditions are established on the spectra of G and on the parameters of G using the inequality of Cauchy Schwarz on the vector space 𝒜.

By the use of the matrix geometric mean #, the matrix Cauchy–Schwarz inequality is given as Y∗X ≤ X∗X # U∗Y∗Y U for k × n matrices X and Y , where Y∗X = U|Y∗X| is a polar decomposition of Y∗X with unitary U. In this note, we generalize Riccati equation as follows: X∗A†X = B for positive semidefinite matrices, where A† is the Moore–Penrose generalized inverse of A. We consider when the matrix geometric mean A # B is a positive semidefinite solution of XA†X = B. For this, we discuss the case where the equality holds in the matrix Cauchy–Schwarz inequality.

In this paper, we first present a generalization of the Cauchy-Schwarz inequality. As an application of our result, we obtain a new sufficient condition for the stability of a class of nonlinear impulsive control systems. We end up this note with a numerical example which shows the effectiveness of our method.

We introduce a concept of Kadison-Schwarz divisible dynamical maps. It turns out that it is a natural generalization of the well known CP-divisibility which characterizes quantum Markovian evolution. It is proved that Kadison-Schwarz divisible maps are fully characterized in terms of time-local dissipative generators. Simple qubit evolution illustrates the concept., Comment: 9 pages

This paper presents a new solution for statistical fusion of multi-sensor information acquired from different fields of view, in a centralized sensor network. The focus is on applications that involve tracking unknown number of objects with time-varying states. Our solution is a track-to-track fusion method in which the information contents of posteriors are combined. Existing information-theoretic solutions for track-to-track fusion in sensor networks are commonly devised based on minimizing the average information divergence from the local posteriors to the fused one. A common approach is to use Generalized Covariance Intersection rule for sensor fusion. This approach works best when all the sensors detect the same object(s), and performs poorly when fields-of-view are different. We suggest Cauchy–Schwarz divergence to be used for measuring information divergence. We demonstrate that employing Cauchy–Schwarz divergence leads to fusion rules that are generally more tolerant to imperfect consensus. We show that the proposed fusion rule for multiple Poisson posteriors is the weighted arithmetic mean of the Poisson densities. Furthermore, we derive the fusion rule for labeled multi Bernoulli filter by approximating the labeled multi Bernoulli density to its first order moment. Numerical experiments show the superior performance of our solution compared to Kullback–Leibler averaging method.