Explicit bounds on some new nonlinear integral inequalities with delay

The purpose of the present note is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions and generalize some results of Li et al. [Some new delay integral inequalities and their applications, J. Comput. Appl. Math. 180 (2005) 191–200]. The inequalities given here can be used to investigate the qualitative properties of certain delay differential equations and delay integral equations.     

A remark on a class of double inequalities of Batir

We prove a class of double inequalities for the gamma function which were conjectured by Batir [On some properties of the gamma function, Expo. Math. 26 (2008) 187–196].     

Extremal matrices in certain determinantal inequalities

The extremal matrices in certain inequalities for determinants of sums are characterized. Related determinantal inequalities involving Hadamard products of positive definite matrices are presented. These inequalities are easy consequences of majorization results recently obtained by Ando and Visick.     

Cone orderings,group majorizations and similarly separable vectors

We generalize some results on majorization in papers by Wu and Debnath [S. Wu, L. Debnath, Inequalities for convex sequences and their applications, Comput. Math. Appl. 54 (2007) 525–534] and Marshall et al. [A.W. Marshall, I. Olkin, F. Proschan, Monotonicity of ratios of means and other applications of majorization, in: O. Shisha (Ed.), Inequalities, Academic Press, New York, 1967, pp. 177–190]. We present sufficient conditions for some vector inequalities to hold in the case of cone orderings and group-induced cone orderings. The framework used is based on results for similarly separable vectors given in paper [M. Niezgoda, Bifractional inequalities and convex cones, Discrete Math. 306 (2006) 231–243].     

An extension of a result of Hwang and Pyo

In this paper, we extend a majorization result of Hwang and Pyo [LAA 332-334 (2001) pp. 15-21] from the ordinary majorization ordering to the class of group induced cone orderings induced by non-effective groups. The case of effective groups is also investigated. A particular attention is paid to positive operators.     

About some new integral inequalities of Wendroff type for discontinuous functions

In this article, we obtain some new nonlinear integral inequalities for discontinuous functions of two independent variables (Wendroff type) by including also inequalities with delay. We deduce new generalizations of earlier results given by R.P. Agarwal, R. Bellman, I. Bihari, B.K. Bondge, V. Lakshmikantham, S. Leela, B.G. Pachpatte for continuous and discrete functions. Furthermore, generalizations of some results for integro-sum inequalities are obtained as well.     

Singularity, Wielandt’s lemma and singular values

In this study, some upper and lower bounds for singular values of a general complex matrix are investigated, according to singularity and Wielandt’s lemma of matrices. Especially, some relationships between the singular values of the matrix A and its block norm matrix are established. Based on these relationships, one may obtain the effective estimates for the singular values of large matrices by using the lower dimension norm matrices. In addition, a small error in Piazza (2002) [G. Piazza, T. Politi, An upper bound for the condition number of a matrix in spectral norm, J. Comput. Appl. Math. 143 (1) (2002) 141-144] is also corrected. Some numerical experiments on saddle point problems show that these results are simple and sharp under suitable conditions.     

Some new integral inequalities of Bellman–Bihari type with delay for discontinuous functions

In this article we investigate some integral functional inequalities of Bellman–Bihari type for piecewise-continuous functions with some fixed points of discontinuity. We also prove a new analogy and generalization of results which were obtained by Bellman and Bihari to integro-sum inequalities with delay and discontinuities that do not belong to Lipschitz’s type.     

Two-sided inequalities for the Struve and Lommel functions

Abstract

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial di?erential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function ₁F₂(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function ₁F₂(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.     

On some new nonlinear discrete inequalities and their applications

In this paper, some new discrete inequalities in two independent variables which provide explicit bounds on unknown functions are established. The inequalities given here can be used as tools in the qualitative theory of certain finite difference equations.     

Singular values of matrix exponentials ∗

The singular values of a matrix and those of its exponential are related via multiplicative majorization. Matrices giving some equalities in the majorization are characterized. As an application, a scalar inequality for the exponential function is generalized to a matrix-valued inequality and the case of equality is examined.     

A Cauchy-Khinchin integral inequality

This paper discusses some Cauchy-Khinchin integral inequalities. Khinchin [2] obtained an inequality relating the row and column sums of 0-1 matrices in the course of his work on number theory. As pointed out by van Dam [6], Khinchin’s inequality can be viewed as a generalization of the classical Cauchy inequality. Van Dam went on to derive analogs of Khinchin’s inequality for arbitrary matrices. We carry this work forward, first by proving even more than general matrix results, and then by formulating them in a way that allows us to apply limiting arguments to create new integral inequalities for functions of two variables. These integral inequalities can be interpreted as giving information about conditional expectations.     

On integral inequalities of the Sobolev type

Summary In this paper some new integral inequalities of the Sobolev type involving many functions of many variables are established. These in turn can be used to serve as generators of other integral inequalities.     

矩阵Frobenius范数不等式

1 引言与引理 矩阵范数与矩阵奇异值问题是数值代数的重要课题,并在矩阵扰动分析,数值计算等分支中起着重要作用.国内外学者对此已作了大量研究.     

About new analogies of Gronwall–Bellman–Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems

In this article we present new integral Gronwall–Bellman–Bihari type inequalities for discontinuous functions (integro-sum inequalities). As applications, we investigate estimated solutions for impulsive differential systems, conditions of boundedness, stability, practical stability.     

Generalized Hardy,Copson, Leindler and Bennett inequalities on time scales

In this paper, we prove some new dynamic inequalities on time scales using Hölder's inequality and Keller's chain rule on time scales. These inequalities, as special cases when the time scale and when , contain some generalizations of integral and discrete inequalities due to Hardy, Copson, Leindler and Bennett.     

Some new rich subspaces of C. Applications

In this paper, we are interested in a class of subspaces of C, introduced by Bourgain [Studia Math. 77 (1984) 245-253]. Wojtaszczyk called them rich in his monograph [Banach Spaces for Analysts, Cambridge Univ. Press, 1991]. We give some new examples of such spaces: this allows us to recover previous results of Godefroy-Saab and Kysliakov on spaces with reflexive annihilator in a very simple way. We construct some other examples of rich spaces, hence having property (V) of Pe?czyński and Dunford-Pettis property. We also recover the results due to Bourgain and Saccone saying that spaces of uniformly convergent Fourier series share these properties, by only using the main result of [Studia Math. 77 (1984) 245-253] and some very elementary arguments. We generalize too these results.     

On the Durrmeyer-type modification of some discrete approximation operators

In [10], for continuous functionsf from the domain of certain discrete operatorsL _n the inequalities are proved concerning the modulus of continuity ofL _nf. Here we present analogues of the results obtained for the Durrmeyer-type modification $\tilde L_n $ ofL _n. Moreover, we give the estimates of the rate of convergence of $\tilde L_n f$ in Hölder-type norms     

On some eigenvalue inequalities for Schatten–von Neumann operators

Michael Gil' recently obtained some bounds for eigenvalues in [J. Funct. Anal. 267 (2014) 3500–3506] and [Commun. Contemp. Math. 18 (2016) 1550022], which improve some classical results related to this aspect. We revisit these results by providing genuinely different arguments (e.g., using Aluthge transform, majorization). New results are derived along our discussions.