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.     

Reverses of the continuous triangle inequality for Bochner integral in complex Hilbert spaces

Some reverses of the continuous triangle inequality for Bochner integral of vector-valued functions in complex Hilbert spaces are given. Applications for complex-valued functions are provided as well.     

A Cauchy-Schwarz type inequality for bilinear integrals on positive measures

If is Borel measurable, define for -finite positive Borel measures on the bilinear integral expression

We give conditions on such that there is a constant , independent of and , with

Our results apply to a much larger class of functions than known before.

     

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Let be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature . If is a compactly supported function of bounded variation on , then satisfies the Sobolev inequality

Conversely, letting be the characteristic function of a domain recovers the sharp form of the isoperimetric inequality for simply connected surfaces with . Therefore this is the Sobolev inequality ``equivalent' to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces.

Under the same assumptions on , if is a closed curve and is the winding number of about , then the Sobolev inequality implies

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature .

     

B-convergence of the implicit midpoint rule and the trapezoidal rule

We present upper bounds for the global discretization error of the implicit midpoint rule and the trapezoidal rule for the case of arbitrary variable stepsizes. Specializing our results for the case of constant stepsizes they easily prove second order optimal B-convergence for both methods.1980 AMS Subject Classification: 65L05, 65L20.     

The error term in Nevanlinna's inequality

An upper bound is given for the error term S(r, {aj}, f) in Nevanlinna's inequality. For given positive increasing functions p and φ with . We prove thatholds, with a small exceptional set of r, for any finite set of points {aj} in the extended plane and any meromorphic function f such that Ψ(T(r, f))=O(P(r)). This improves the known results of A. Hinkkanen and Y. F. Wang. The sharpness of the estimate is also considered.     

A retarded integral inequality and its applications

We prove a retarded nonlinear integral inequality and present some applications of it to the global existence of solutions to differential equations with time delay.     

An integral recursive inequality and applications

An integral recursive inequality for two functions is obtained. It is used to describe the equality cases in the related inequalities. The applications involve some bi-Hermitian forms, integral transformations, and confluent hypergeometric functions.      

A logarithmic Hardy inequality

We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden-Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved by radial functions, in some range of the parameters.     

Multidimensional weighted Opial inequalities

In this article we produce Opial-type weighted multidimensional inequalities over balls and arbitrary smooth bounded domains. The inequalities are sharp. The functions under consideration vanish on the boundary.     

A Cauchy--Khinchin--van Dam matrix inequality

We derive a matrix inequality, which generalizes the Cauchy inequality for vectors, Khinchin's inequality for zero-one matrices and van Dam's inequality for matrices.     

Generalization of a sharp Hölder's inequality and its application

In this paper, we generalize Hu Ke's sharpness of Hölder's inequality. As application, the obtained result is used to improve the well-known Opial-Olech inequality.     

A generalized Gronwall inequality and its application to a fractional differential equation

This paper presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation.     

Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval

The problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well-known metric, which preserves the expected interval of the fuzzy number, is completely solved. The previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number. Properties of this new trapezoidal approximation operator are studied.     

Extension of Hilbert's inequality

In this paper we make some further extensions of discrete Hilbert's inequality by using Euler-Maclaurin summation formula. We give the improvements of some previously obtained results and also compare our results with some previously known from the literature.     

Complementary inequalities to improved AM-GM inequality

Following an idea of Lin, we prove that if A and B are two positive operators such that 0 mI ≤ A ≤m'I≤ M'I ≤ B ≤ MI, then Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2Φ~2(A≠B) and Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2(Φ(A)≠Φ(B))~2 where K(h)=(h+1)~2/4 and h = M/m and Φ is a positive unital linear map.     

Sharp error estimates for the numerical differentiation formulas on the classes of entire functions of exponential type

We establish sharp error estimates for some numerical di.erentiation formulas on the classes of entire functions of exponential type. The estimates strengthen some classical sharp inequalities of approximation theory.     

Inverse inequality estimates with symbolic computation

In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as accurately as possible. We apply symbolic computation methods to the situation of square elements and are able to improve the previously known upper bound, given in “p- and hp-finite element methods” (Schwab, 1998), by a factor of 8. More precisely, we try to evaluate the corresponding determinant using the holonomic ansatz, which is a powerful tool for dealing with determinants, proposed by Zeilberger in 2007. However, it turns out that this method does not succeed on the problem at hand. As a solution we present a variation of the original holonomic ansatz that is applicable to a larger class of determinants, including the one we are dealing with here. We obtain an explicit closed form for the determinant, whose special form enables us to derive new and tight upper resp. lower bounds on the maximal eigenvalue, as well as its asymptotic behaviour.     

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.