Two-weight weak-type inequalities for potential type operators

For the potential type operator
TФf（x）=∫RnФ（x-y）f（y）dy,
where Ф is a non-negative locally integrable function on R^n and satisfies weak growth condition, a two-weight weak-type （p,q） inequality for TФ is obtained.     

On the embedding of anisotropic Nikol’skiĭ-Besov mixed norm spaces

The embedding of the anisotropic spaces $B_{p_1 , \ldots ,p_n ,\theta }^{\omega _1 , \ldots ,\omega _n } \left( {\mathbb{R}^n } \right)$ with mixed norm is studied. We establish some necessary and sufficient conditions of the embedding $B_{p_1 , \ldots ,p_n ,\theta }^{\omega _1 , \ldots ,\omega _n } \left( {\mathbb{R}^n } \right) \subset L^{q_1 , \ldots ,q_n } \left( {\mathbb{R}^n } \right)$ .     

On some general Hilbert’s inequalities with non-conjugate parameters

A generalized discrete Hilbert’s and Hardy-Hilbert’s inequality with non-conjugate parameters can be established by means of Euler-Maclaurin summation formula. We derive some general results for homogeneous functions and compare our results with some previously known from the literature. We also obtain the improvements on some earlier results.     

Weighted fractional Bernstein’s inequalities and their applications

This paper studies the weighted, fractional Bernstein inequality for spherical polynomials on S^d-1\(\left( {0.1} \right)\;{\left\| {{{\left( { - {\Delta _0}} \right)}^{{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}}f} \right\|_{p,w}} \leqslant {C_w}{n^r}{\left\| f \right\|_{p,w}}\;for\;all\;f \in \Pi _n^d\), where Π_n^d denotes the space of all spherical polynomials of degree at most n on S^d-1 and (-Δ₀)^r/2 is the fractional Laplacian-Beltrami operator on S^d-1. A new class of doubling weights with conditions weaker than the A_p condition is introduced and used to characterize completely those doubling weights w on S^d-1 for which the weighted Bernstein inequality (0.1) holds for some 1 ≤ p ≤ 8 and all r > t. It is shown that in the unweighted case, if 0 < p < 8 and r > 0 is not an even integer, (0.1) with w = 1 holds if and only if r > (d - 1)((1/p) - 1). As applications, we show that every function f ∈ L_p(S^d-1) with 0 < p < 1 can be approximated by the de la Vallée Poussin means of a Fourier-Laplace series and establish a sharp Sobolev type embedding theorem for the weighted Besov spaces with respect to general doubling weights.     

On symmetrized stochastic convexity and the inequalities of Hermite–Hadamard type

Aequationes mathematicae - In this paper the concept of symmetrized convex stochastic processes is introduced. Some characterizations involving Hermite–Hadamard type inequalities and a...     

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.     

On Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Pólya and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemannian manifolds and derive the well-known Taikov and Hardy-Littlewood-Pólya inequalities for functions defined on the d-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of another class. In addition, we establish sharp Solyar type inequalities for unbounded closed operators with closed range.

     

On Hardy－Hilbert‘s Integral Inequality and Its Equivalent Form

In this paper, by introducing three parameters a, b and λ, we give some new generalizations of Hardy-Hilbert‘s integral inequality. As applications, we con-sider its equivalent form and some particular results.     

New Grüss’ type inequalities for functions of bounded variation and applications

Some new sharp Grüss’ type inequalities for functions of bounded variation and applications for selfadjoint operators in Hilbert spaces are given.     

On the mean curvature of spacelike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein’s type problems

Several uniqueness results for the spacelike slices in certain Robertson–Walker spacetimes are proved under boundedness assumptions either on the mean curvature function of the spacelike surface or on the restriction of the time coordinate on the surface when the mean curvature is a constant. In the nonparametric case, a uniqueness result and a nonexistence one are proved for bounded entire solutions of some constant mean curvature spacelike differential equations.     

基本Bernstein多项式的一个积分不等式(英文)

本文研究了Bernstein多项式.通过归纳法,建立了一个与基本Bernstein多项式有关的积分型不等式.     

On weighted Ostrowski type,Trapezoid type,Grüss type and Ostrowski–Grüss like inequalities on time scales

In this paper, we first derive a weighted Montgomery identity on time scales and then establish weighted Ostrowski-type, Trapezoid-type, Grüss-type and Ostrowski–Grüss-like inequalities on time scales, respectively. These results not only provide a generalization of the known results, but also give some other interesting inequalities on time scales as special cases.     

On the stability of general convex programs under slater’s condition and primal solution boundedness

This paper was motivated by an article by Best and Chakravarti, who presented some stability results for convex quadratic programs under linear perturbation of the data. We show that the regularity conditions assumed are much too restrictive and demonstrate that stronger stability results follow under weaker assumptions (primal solution boundedness and the Slater condition) and from known results, not only for convex quadratic problems but for general convex programs with general perturbations. In so doing, we give a simple and reasonably complete characterization of the stability of an important class of well-behaved convex programs, collecting results that heretofore have apparently not been presented in a unified manner. The results, virtually all from Hogan and Robinson, involve mainly stability of the feasible region and solution existence under small perturbations, and continuity and differentiability of the optimal value function. We note that Auslender and Coutat have recently provided similar extensions for saddle points of generalized linear-quadratic programs introduced by Rockafellar and Wets, utilizing the same assumptions that we use in this paper     

On the rate of convergence of Krasnosel’skiĭ-Mann iterations and their connection with sums of Bernoullis

In this paper we establish an estimate for the rate of convergence of the Krasnosel’ski?-Mann iteration for computing fixed points of non-expansive maps. Our main result settles the Baillon-Bruck conjecture [3] on the asymptotic regularity of this iteration. The proof proceeds by establishing a connection between these iterates and a stochastic process involving sums of non-homogeneous Bernoulli trials. We also exploit a new Hoeffdingtype inequality to majorize the expected value of a convex function of these sums using Poisson distributions.