On certain integral inequalities related to Opial's inequality

The aim of the present paper is to establish some new integral inequalities involving three functions and their derivatives which in the special cases yield the well known Opial inequality and some of its generalizations.     

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.     

On two inequalities similar to Opial's inequality in two independent variables

In the present paper we establish two new integral inequalities similar to Opial's inequality in two independent variables. The inequalities established in this paper are similar to the analogues of Calvert's generalizations of Opial's inequality, in two independent variables and contains in the special case the analogue of Opial's inequality given by G. S. Yang in two independent variables.     

A fractional differential inequality with application

We establish a fractional differential inequality using desingularization techniques combined with some generalizations of algebraic Bihari-type inequalities. We use this inequality to prove global existence and determine the asymptotic behavior of solutions for a family of fractional differential equations.     

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.     

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.     

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.     

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 the identric and logarithmic means

Summary Leta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(b ^b /a ^a ) ^1/(b–a) , fora b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b – a)/(logb – loga), fora b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) = . In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means.     

Integral inequalities related to Hardy's inequality

Some generalizations are given of Hardy's inequality relating toL ^p -spaces. The results include many existing integral inequalities.Dedicated to Professor Janos Aczél on his 60th birthday     

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.     

On discrete inequalities of the Poincaré type

The aim of the present paper is to establish some new discrete inequalities of the Poincaré type involving functions ofn independent variables and their first order forward differences. The proofs given here are quite elementary and our results provide new estimates on this type of discrete inequalities.     

若干有关欧阳不等式的非线性积分不等式和离散不等式

获得几个非线性的积分不等式及离散不等式．它们和近期Ｐａｃｈｐａｔｅ［１］中欧阳亮不等式［２］的推广有关．作为特殊情形还导出了一些具有幂非线性的新不等式．为说明结果的有用性，讨论了某个非线性差分方程解的有界性．     

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.     

A note on certain integral inequalities with delay

In this paper we establish some new integral inequalities with delay, which can be used as tools in the theory of some new classes of differential and integral equations. An application to obtain a bound on the solution of a certain integral equation is also given.     

Some new nonlinear difference inequalities and their applications

In this paper, we establish some new nonlinear difference inequalities in two independent variables, which can be used as handy tools in the study of qualitative properties of solutions of certain classes of difference equations.     

On Opial-type integral inequalities

The aim of the paper is to establish some new integral inequalities involving two functions and their first order and higher order derivatives. Our results in the special cases yield the well-known Opial inequality and some of its generalizations.     

Majorization and relative concavity

In this paper, some results established in [H.-N. Shi, Refinement and generalization of a class of inequalities for symmetric functions, Math. Practice Theory 29 (4) (1999) pp. 81-84] are extended from the classical majorization preordering to group-induced cone orderings. To this end the notion of relative concavity introduced in [C.P. Niculescu, F. Popovici, The extension of majorization inequalities within the framework of relative convexity, J. Inequal. Pure Appl. Math. 7 (1) (2005) (Article 27)] is used. In addition, some Ky Fan’s inequalities are discussed.     

Newman不等式和Klamkin不等式的指数推广

Exponential generalizations of Newman's inequality and Klamkin's inequality are established by the Wang Wan-lan's inequality, and they are extended to the cases involving general elementary symmetric functions. As an application, some new inequalities for a simplex are established. In addition, an open problem is posed.     

On CLARKSON's Inequalities and Interpolation

We consider some elementary proofs of local versions of CLARKSON 's inequalities and point out the fact that these inequalities can be generalized to hold for a much wider class of parameters. In particular it is easy to generalize our interpolation proof in various ways to higher dimensions. We point out explicitely some examples of such generalizations and we also prove some corresponding global versions. In this elementary way we obtain both new proofs of some previous results of this kind and also some new complements, unifications and further generalizations of these results.