Some Refinement of Holder’s and Its Reverse Inequality

Holder’s inequality, its refinement, and reverse have received considerable attention in the theory of mathematical analysis and differential equations. In this paper, we give some refinements of Holder’s inequality and its reverse using a simple analytical technique of algebra and calculus. Our results show many results related to holder’s inequality as special cases of the inequalities presented.

Necessary and Sufficient Conditions of Doubly Weighted Hardy-Littlewood-Sobolev Inequality

Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.

Some Equivalent Forms of Bernoulli’s Inequality: A Survey

The main purpose of this paper is to link some known inequalities which are equivalent to Bernoulli’s inequality.

On Hardy-type integral inequalities

The Hardy integral inequality is one of the most important inequalities in analysis. The present paper establishes some new Copson-Pachpatte （C-P） type inequalities, which are the generalizations of the Hardy integral inequalities on binary functions.

Modified approximation and error estimation for King’s type(p,q)-BBH operators

In this paper,the King’s type modification of(p,q)-Bleimann-Butzer and Hahn operators is defined.Some results based on Korovkin’s approximation theorem for these new operators are studied.With the help of modulus of continuity and the Lipschitz type maximal functions,the rate of convergence for these new operators are obtained.It is shown that the King’s type modification have better rate of convergence,flexibility than classical(p,q)-BBH operators on some subintervals.Further,for comparisons of the operators,we presented some graphical examples and the error estimation in the form of tables through MATLAB(R2015a)

Commutators of Singular Integral Operators Related to Magnetic Schrdinger Operators

Let A=-（ -ia）·（ -ia）＋V be a magnetic Schrodinger operator on L^2（R^n）,n〉2,where a：=（a1,…,an）∈Lloc^2（R^n,R^n） and 0≤V ∈Lloc^1（R^n）.In this paper, we show that for a function b in Lipschitz space Lipa（R^n）with a∈（0,1）,the commutator [b,V1/2A^-1/2]is bounded from L^p（R^n）to L^q（R^n）,where p,q∈（1,2] and 1/p-1/q=a/n.

REGULARITY FOR VERY WEAK SOLUTIONS TO A-HARMONIC EQUATION

In this paper, the following result is given by using Hodge decomposition and weak reverse Holder inequality： For every r1 with P-（2^n＋1 100^n^2 p（2^3＋n/（P-1）＋1）b/a）^-1〈r1〈p,there exists the exponent r2 = r2（n, r1,p） 〉 p, such that for every very weak solution u∈W^1r1,loc（Ω） to A-harmonic equation, u also belongs to W^1r2,loc（Ω） . In particular, u is the weak solution to A-harmonic equation in the usual sense.

Some General Inequalities for Choquet Integral

With the development of fuzzy measure theory, the integral inequalities based on Sugeno integral are extensively investigated. We concern on the inequalities of Choquuet integral. The main purpose of this paper is to prove the H?lder inequality for any arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions f, g and h are comonotone, and there are three weights. Then we prove Minkowski inequality and Lyapunov inequality for Choquet integral. Moreover, when any two of these integrated functions f1, f2, …, fn are comonotone, we also obtain the Hölder inequality, Minkowski inequality and Lyapunov inequality hold for Choquet integral.

Hermite-Hadamard Type Fractional Integral Inequalities for Preinvex Functions

In this article, we extend some estimates of the right-hand side of the Hermite-Hadamard type inequality for preinvex functions with fractional integral.The notion of logarithmically s-Godunova-Levin-preinvex function in second sense is introduced and then a new Herrnite-Hadarnard inequality is derived for the class of logarithmically s-Godunova-Levin-preinvex function.

Some Hermite-Hadamard Type Inequalities for Differentiable Co-Ordinated Convex Functions and Applications

In this paper, we shall establish an inequality for differentiable co-ordinated convex functions on a rectangle from the plane. It is connected with the left side and right side of extended Hermite-Hadamard inequality in two variables. In addition, six other inequalities are derived from it for some refinements. Finally, this paper shows some examples that these inequalities are able to be applied to some special means.

The Generalization on Inequalities of Hermite-Hadamard's Integration

Some new inequalities of Hermite-Hadamard's integration are established. As for as inequalities about the righthand side of the classical Hermite-Hadamard's integral inequality refined by S Qaisar in [3], a new upper bound is given. Under special conditions,the bound is smaller than that in [3].

Regularity for weakly(K_(1),K_(2))-quasiregular mappings

In this paper, we first give the definition of weakly (K1,K2)-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Holder inequality, we obtain their regularity property: For any q1 that satisfies 0＜K1n(n+4)/22n+1×100n2[23n/2(25n+1)](n - q1) ＜ 1, there exists p1 = p1(n,q1,K1,K2)＞n, such that any (K1,K2)-quasiregular mapping f ∈ W1,q1loc(Ω,Rn) is in fact in W1n,p1loc (Ω, Rn). That is, f is (K1, K2)-quasiregular in the usual sense.

Regularity Results for the Generalized Beltrami System

For the generalized Beltrami system with two characteristic matrices, we deal with the regularity of its very weak solutions in the Sobolev class (1 < r < n). By changing the generalized Beltrami system into a class of a divergent elliptic system with nonhomogeneous items, we obtain that each of its very weak solutions is essentially a classical weak solution of a usual Sobolev class. Furthermore, we also establish a higher regularity of its weak solution if the regularity hypotheses of two characteristic matrices are improved.

Regularity for Quasi-linear Degenerate Elliptic Equations with VMO Coefficients

In this paper we establish an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coefficient matrix A（x, u） satisfies a VMO condition in the variable x uniformly with respect to all u, and the lower order item B（x, u, △↓u） satisfies the subcritical growth （1.2）. In particular, when F（x） ∈ L^q（Ω） and f（x） ∈ L^γ（Ω） with q,γ 〉 for any 1 〈 p 〈 ＋∞, we obtain interior HSlder continuity of any weak solution of （1.1） u with an index κ = min{1 - n/q, 1 - n/γ}.

Spectrum and Singular Integrals on a New Weighted Function Space

We introduce a new function space that is much finer than the classical Lebesgue spaces, and investigate the continuity and the discontinuity of the Calderon-Zygmund operators on the new weighted spaces. In order to identify the continuity of singular integrals, we prove an interpolation theorem on the new spaces and propose a concept of the spectrum of base functions.

Some Source Coding Theorems and 1:1 Coding Based on Generalized Inaccuracy Measure of Order α and Typeβ

In this paper,we have established some noiseless coding theorems for a generalized parametric‘useful’inaccuracy measure of orderαand typeβand generalized mean codeword length.Further,lower bounds on exponentiated useful code length for the best 1:1 code have been obtained in terms of the useful inaccuracy of orderαand typeβand the generalized average useful codeword length.

NEW DYNAMIC INEQUALITIES FOR DECREASING FUNCTIONS AND THEOREMS OF HIGHER INTEGRABILITY

In this paper we establish some new dynamic inequalities on time scales which contain in particular generalizations of integral and discrete inequalities due to Hardy, Littlewood, Polya, D＇Apuzzo, Sbordone and Popoli. We also apply these inequalities to prove a higher integrability theorem for decreasing functions on time scales.