On discrete and continuous wirtinger inequalities

We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.     

Dynamic Picone's identity and its applications

We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.     

Reconstruction of convex bodies from surface tensors

We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. When only measurements subject to noise of surface tensors are available for reconstruction, we recommend to use certain values of the surface tensors, namely harmonic intrinsic volumes instead of the surface tensors evaluated at the standard basis. The second algorithm we present is based on harmonic intrinsic volumes and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise variables.     

An Improved Hardy's Inequality

In this paper we shall extend Hardy's inequality associated with Fourier transform to the strip n(2-p) ≤ σ < n+p(N +1) where N = [n(1/p-1)], the greatest integer not exceeding n(1/p−1).     

On Pólya–Szegö reverse of Schwarz's inequality

We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.     

Opial's and wirtinger's type discrete inequalities in two independent variables

We develop discrete Taylor's formula in two independent variables and use it to establish Opial's type inequality involving higher order partial differences. This is followed by several variations of discrete Wirtinger's inequality in two independent variables     

On critical cases of Sobolev's inequalities for Heisenberg groups

We prove some Trudinger-type inequalities and Brezis-Gallouet-Wainger inequality on the Heisenberg group, extending to this context the Euclidean results by T. Ozawa.     

On the best constant in Gaffney inequality

We discuss the value of the best constant in Gaffney inequality namely

${6\mathrm{?}\omega 6}_{L^2}^2\le C({6d\omega 6}_{L^2}^2+{6\delta \omega 6}_{L^2}^2+{6\omega 6}_{L^2}^2)$

when either $\nu \wedge \omega =0$ or $\nu ?\omega =0$ on ?Ω.     

On an upper bound of Sanov's inequality for the probability of a large deviation

In this paper a sharp upper bound of Sanov's inequality for the large probability of large deviations under the multinomial distribution is obtained. Examples of computing the upper bounds for the probabilities of misclassification in discriminant analysis are also given.     

Refinements of the Cauchy–Bunyakovsky–Schwarz inequality for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous functions of selfadjoint operators in Hilbert spaces that improve the Cauchy–Bunyakovsky–Schwarz inequality, are given.