Some results on extending and sharpening the Weierstrass product inequalities

In this paper, we establish two extensions of Weierstrass's inequality involving symmetric functions by means of the theory of majorization, and give an interesting sharpness of Weierstrass's inequality by using the arithmetic-geometric mean inequality. Furthermore, we apply these results to improve a well-known inequality and deduce some new inequalities.     

Dynamic approach to a stochastic domination: The FKG and Brascamp-Lieb inequalities

A coupling based on a pair of stochastic differential equations is introduced to show a stochastic domination for a system with continuous spins, from which the FKG and Brascamp-Lieb like inequalities follow.

     

Some matrix inequalities related to Kwong functions

The aim of this note is to investigate some inequalities related to Kwong functions. Refinements of Kaur’s and Zhan’s inequality are presented by the Hadamard product. In addition, variations of the Heinz–Heron type means on Kwong functions are also given.     

Transport-majorization to analytic and geometric inequalities

We introduce a transport-majorization argument that establishes a majorization in the convex order between two densities, based on control of the gradient of a transportation map between them. As applications, we give elementary derivations of some delicate Fourier analytic inequalities, which in turn yield geometric “slicing-inequalities” in both continuous and discrete settings. As a further consequence of our investigation we prove that any strongly log-concave probability density majorizes the Gaussian density and thus the Gaussian density maximizes the Rényi and Tsallis entropies of all orders among all strongly log-concave densities.     

Some inequalities in inner product spaces related to the generalized triangle inequality

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.     

Some results related to the Corach-Porta-Recht inequality

Let be the algebra of all bounded operators on a complex Hilbert space and let be an invertible self-adjoint (or skew-symmetric) operator of . Corach-Porta-Recht proved that

The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht Inequality; (ii) a necessary condition (resp. necessary and sufficient condition, when for the invertible positive operators to satisfy the operator-norm inequality for all in ; (iii) a necessary and sufficient condition for the invertible operator in to satisfy

     

Some covariance inequalities in Wiener space

In this work we give an account of some covariance inequalities in abstract Wiener space. An FKG inequality is obtained with positivity and monotonicity being defined in terms of a given cone in the underlying Cameron-Martin space. The last part is dedicated to convex and log-concave functionals, including a proof of the Gaussian conjecture for a particular class of log-concave Wiener functionals.     

Hardy inequalities related to Grushin type operators

We prove some Hardy type inequalities related to the Grushin type operator .

     

Some inequalities related to Sobolev norms

In this paper, we study some properties related to the new characterizations of Sobolev spaces introduced in Bourgain and Nguyen (C R Acad Sci, 343:75?C80, [2006]), Nguyen (J Funct Anal 237: 689?C720, [2006]; J Eur Math Soc 10:191?C229, [2008]). More precisely, we establish variants of the Poincaré inequality, the Sobolev inequality, and the Rellich?CKondrachov compactness theorem, where ${\int_{\mathbb{R}^N} |\nabla g|^p \;dx}$ is replaced by some quantity of the type $$I_{\delta} (g) =\mathop{\int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N}}_{|g(x) - g(y)| > \delta}\frac{\delta^p}{|x-y|^{N+p}}\, dx \, dy.$$      

Some Hardy type integral inequalities

We present new kinds of Hardy integral inequalities involving some generalization and improvement.     

Some inequalities related to the Stam inequality

Zamir showed in 1998 that the Stam classical inequality for the Fisher information (about a location parameter)

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.

     

Some Bonnesen-style inequalities for higher dimensions

We discuss the higher dimensional Bonnesen-style inequalities.Though there are many Bonnesen-style inequalities for domains in the Euclidean plane R2 few results for general domain in R n(n ≥ 3) are known.The results obtained in this paper are for general domains,convex or non-convex,in Rn.     

Some inequalities for absolute moments

In this note, we obtain several inequalities for absolute moments of sums and differences of independent random variables, using one representation of absolute moments in terms of the characteristic function and inequalities for characteristic functions.     

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient ? _?? = (? _x ,|x| ^?? ? _y ) by a direct and simple approach. Moreover, similar method is applied to obtain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L ^p . We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.