Journal of Inequalities and Applications

List of Papers (Total 3,406)

Carlson type inequalities

A scale of Carlson type inequalities are proved and the best constants are found. Some multidimensional versions of these inequalities are also proved and it is pointed out that also a well-known inequality by Beurling–Kjellberg is included as an endpoint case.

A new method for nonsmooth convex optimization

A new method for minimizing a proper closed convex function is proposed and its convergence properties are studied. The convergence rate depends on both the growth speed off at minimizers and the choice of proximal parameters. An application of the method extends the corresponding results given by Kort and Bertsekas for proximal minimization algorithms to the case in which the...

A characterization of chaotic order and a problem

In our previous notes, we give a useful characterization of the chaotic order, i.e., for positive invertible operators and . In this note, we present a short proof to the characterization of the chaotic order and give an answer to a related problem on it. Moreover we consider the orders defined by as an interpolation between the chaotic order and the usual order via the Furuta...

On some inequalities and stability results related to the exponential function

Some inequalities related to the exponential function are solved and the stability of the functional equations and is studied.

Some embeddings of weighted sobolev spaces on finite measure and quasibounded domains

We show that several of the classical Sobolev embedding theorems extend in the case of weighted Sobolev spaces to a class of quasibounded domains which properly include all bounded or finite measure domains when the weights have an arbitrarily weak singularity or degeneracy at the boundary. Sharper results are also shown to hold when the domain satisfies an integrability...

Generalized Gram–Hadamard inequality

We generalize the classical Gram determinant inequality. Our generalization follows from the boundedness of the antisymmetric tensor product operator. We use fermionic Fock space methods.

On the comparison of trigonometric convolution operators with their discrete analogues for Riemann integrable functions

This note is concerned with a comparison of the approximation-theoretical behaviour of trigonometric convolution processes and their discrete analogues. To be more specific, for continuous functions it is a well-known fact that under suitable conditions the relevant uniform errors are indeed equivalent, apart from constants. It is the purpose of this note to extend the matter to...

A refinement of various mean inequalities

A new refinement of the classical arithmetic mean and geometric mean inequality is given. Moreover, a new interpretation of the classical mean is given and this refinement theorem is generalized.

Two polynomial division inequalities in

This paper is a first attempt to give numerical values for constants and , in classical estimates and where is an algebraic polynomial of degree at most and denotes the -metric on . The basic tools are Markov and Bernstein inequalities.

On Ozeki's inequality

We shall obtain the best bound in Ozeki's inequality which estimates the difference of Cauchy's inequality. We also give an operator version of Ozeki's inequality which extends an inequality on the variance of an operator.

An inequality on solutions of heat equation

Let be the solution of the initial value problem for the dimensional heat equation. Then, for any and for any , an inequality about and is obtained.

An integral analogue of the Ostrowski inequality

We give an integral analogue of the Ostrowski inequality and several extensions, allowing in particular for multiple linear constraints.

On generalized Wazewski and Lozinskii inequalities for semilinear abstract differential-delay equations

A semilinear abstract differential-delay equation with a nonautonomous linear part is considered. Solution estimates are derived. They generalize Wazewski and Lozinskii inequalities. Conditions for global stability are established.

Von Neumann–Jordan constant for Lebesgue–Bochner spaces

The von Neumann–Jordan (NJ-) constant for Lebesgue–Bochner spaces is determined under some conditions on a Banach space . In particular the NJ-constant for as well as (the space of -Schatten class operators) is determined. For a general Banach space we estimate the NJ-constant of , which may be regarded as a sharpened result of a previous one concerning the uniform non-squareness...

Five integral inequalities; an inheritance from Hardy and Littlewood

This paper is concerned with five integral inequalities considered as generalisations of an inequality first discovered by G.H. Hardy and J.E. Littlewood in 1932. Subsequently the inequality was considered in greater detail in the now classic text Inequalities of 1934, written by Hardy and Littlewood together with G. Pólya. All these inequalities involve Lebesgue square...

Generalized variational inequalities of the Hartman–Stampacchia–Browder type

We obtain several generalized variational inequalities from an equilibrium theorem due to the first author under more weaker hypothesis and in more general setting than known ones. Our new results extend, unify and improve many known Hartman–Stampacchia–Browder type variational inequalities for u.s.c. or monotone type multimaps. Our proofs are also much simpler than known ones.

Characterizations of chaotic order via generalized Furuta inequality

A characterization of chaotic order is given by using generalized Furuta inequality and its application to related norm inequalities is given as a precise estimation of our previous paper [15]. Also parallel results related to generalized Furuta inequality are given by using nice characterization of chaotic order by Fujii et al. [7].

Relationship between stochastic inequalities and some classical mathematical inequalities

The notions of association and dependence of random variables, rearrangements, and heterogeneity via majorization ordering have proven to be most useful for deriving stochastic inequalities. In this survey article we first show that these notions are closely related to three basic inequalities in classical mathematical analysis: Chebyshev's inequality, the Hardy-Littlewood-Pólya...

On some fractional order hardy inequalities

Weighted inequalities for fractional derivatives ( fractional order Hardy-type inequalities) have recently been proved in [4] and [1]. In this paper, new inequalities of this type are proved and applied. In particular, the general mixed norm case and a general twodimensional weight are considered. Moreover, an Orlicz norm version and a multidimensional fractional order Hardy...

Weighted Lagrange and Hermite–Fejér interpolation on the real line

For a wide class of weights, a systematic investigation of the convergence-divergence behavior of Lagrange interpolation is initiated. A system of nodes with optimal Lebesgue constant is found, and for Hermite weights an exact lower estimate of the norm of projection operators isgiven. In the same spirit, the case of Hermite–Fejér interpolation is also considered.

Embedding variational inequalities and their generalizations into a separation scheme

When the domain of a variational inequality is given as intersection of level sets of functionals, then it is useful to introduce a separation scheme. Starting from this it is possible to carry out, in a uniform way, several topics, as penalization, regularization, gap functions, duality.

Weighted modular inequalities for monotone functions

Weight characterizations of weighted modular inequalities for operators on the cone of monotone functions are given in terms of composition operators on arbitrary non-negative functions with changes in weights. The results extend to modular inequalities, those corresponding to weighted Lebesgue spaces given by E.T. Sawyer [15]. Application to Hardy and fractional integral...

Remarks on the hardy inequality

Let be an open subset of with finite Lebesgue -measure, let be the distance from to the boundary of , and let . We give a simple direct proof that if satisfies the plumpness condition of Martio and Väisälä [10], then the inequality of Hardy type, holds whenever . We also show that the plumpness condition may be replaced by ones. which enable domains with lower-dimensional...

Hermite interpolation and an inequality for entire functions of exponential type

Let . It is shown that if is an entire function of exponential type and , where is a sequence of real numbers satisfying , for , then , where depends only on , and . A sampling theorem for irregularly spaced sample points is obtained as a corollary. Our proof of the main result contains ideas which help us to obtain an extension of a theorem of R.J. Duffin and A.C. Schaeffer...

Small values of polynomials: Cartan, Pólya and others

Let be a monic polynomial of degree , and , . A classic lemma of Cartan asserts that the lemniscate can be covered by balls , whose diameters satisfy For , this shows that has an area at most . Pólya showed in this case that the sharp estimate is . We discuss some of the ramifications of these estimates, as well as some of their close cousins, for example when is normalized to...