Multidimensional Ostrowski inequalities, revisited

New very general multidimensional Ostrowski type inequalities are established, some of them prove to be sharp. They involve the · and ·_p norms of the engaged mixed partial of nth order n1. In establishing them, other important multivariate results of Montgomery type identity are developed and presented for the first time.     

Reduction of Opial-type inequalities to norm inequalities

Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the nonincreasing rearrangement are presented.

     

On weighted Hermite-Hadamard inequalities

In this paper, we give a weighted form of the Hermite-Hadamard inequalities. Some applications of them are also derived. The results presented here would provide extensions of those given in earlier works. Finally we pose two interesting problems.     

On some new integral inequalities of Gronwall-Bellman-Pachpatte type

In this paper, we establish new nonlinear integral inequalities of Gronwall-Bellman-Pachpatte type. These inequalities generalize some former famous inequalities and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of some nonlinear ordinary differential and integral equations. The purpose of this paper is to extend certain results which proved by Pachpatte in [Inequalities for Differential and Integral Equations, Academic Press, New York and London, 1998]. Some applications are also given to illustrate the usefulness of our results.     

Some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables

Several inequalities for differentiable convex, wright-convex and quasi-convex mapping are obtained respectively that are connected with the celebrated Hermite-Hadamard integral inequality. Also, some error estimates for weighted Trapezoid formula and higher moments of random variables are given.     

Poincaré and Opial inequalities for vector-valued convolution products

Various L^p$L^p$ form Poincaré and Opial inequalities are given for vector-valued convolution products. We apply our results to infinitesimal generators of _C0$C_0$-semigroups and cosine functions. Typical examples of these operators are differential operators in Lebesgue spaces.     

Sharp Marchaud and converse inequalities in Orlicz spaces

For spaces on , and , sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on , and ) for which is convex for some , , where is the Orlicz function. Sharp converse inequalities for such spaces are deduced.

     

Opial type inequalities for cosine and sine operator functions

Various L _p form Opial type inequalities are given for cosine and sine operator functions with applications.     

Sharp weighted inequalities for multilinear fractional maximal operators and fractional integrals

In this paper, we study weighted inequalities for multilinear fractional maximal operators and fractional integrals. We prove sharp weighted Lebesgue space estimates for both operators when the vector of weights belongs to . In addition we prove sharp two weight mixed estimates for multilinear operators in the spirit of the linear estimates given in 3 .     

Best constants in the Hardy-Rellich type inequalities on the Heisenberg group

We prove some sharp Hardy-Rellich inequalities on the Heisenberg group, using the method given by D.G. Costa in [D.G. Costa, Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities, J. Math. Anal. Appl. 337 (2008) 311-317].     

Pitt and Boas inequalities for Fourier and Hankel transforms

We prove Pitt and Boas inequalities for products of radial functions and spherical harmonics in Rⁿ${\mathbb{R}}^n$. In the process, we obtain upper and lower estimates of the operator norm of the Hankel transform with power weights. Our inequalities are sharp in some specific cases.     

Higher order Ostrowski type inequalities over Euclidean domains

The classical Ostrowski inequality for functions on intervals estimates the value of the function minus its average in terms of the maximum of its first derivative. This result is extended to functions on general domains using the L^∞ norm of its nth partial derivatives. For radial functions on balls the inequality is sharp.     

On a class of weighted anisotropic Sobolev inequalities

In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities where different derivatives have different weight functions. These inequalities are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider Sobolev inequalities on finite cylinders, the weight being a power of the distance function from the top or the bottom of the cylinder. We also prove similar inequalities in the more general case in which the weight is a power of the distance function from a higher codimension part of the boundary.     

On some sharp weighted norm inequalities

Given a weight ω, we consider the space which coincides with when ω∈A_p. Sharp weighted norm inequalities on for the Calderón-Zygmund and Littlewood-Paley operators are obtained in terms of the A_p characteristic of ω for any 1<p<∞.     

Some generalized integral inequalities and their applications

In this paper, we generalize some integral inequalities to more general situations. We establish bounds on the solutions and, by means of examples, we show the usefulness of our results in investigating the global existence of the solutions of integral equations and differential equations.     

Some complements to the Jensen and Chebyshev inequalities and a problem of W. Walter

Motivated by an integral inequality conjectured by W. Walter, we prove some general integral inequalities on finite intervals of the real line. In addition to supplying new proofs of Walter's conjecture, the general inequalities furnish a reverse Jensen inequality under appropriate conditions and provide generalizations of Chebyshev's integral inequality.     

On weighted inequalities for martingale transform operators

For 1 < p < ∞, the almost surely finiteness of is a necessary and sufficient condition in order to have almost surely convergence of the sequences {E(f|?_n)} with f ∈ L^p(v dP). This condition is also equivalent to have weighted inequalities from L^p(v dP) into L^p(u dP) for some weight u for Doob's maximal function, square function and generalized Burkholder martingale transforms. Similarly, E(u|?₁) < ∞ turns out to be necessary and sufficient for the above weighted inequalities to hold for some v.