In this paper, we introduce the notion of sesquilinear map on Β(H) . Based on this notion, we define the quadratic map, which is the generalization of positive linear map. With the help of this concept, we prove several well-known equality and inequality...  

We state, prove, and discuss new general inequality for convex and increasing functions. As a special case of that general result, we obtain new fractional inequalities involving fractional integrals and derivatives of Riemann-Liouville type. Consequently, we get the inequality of H. G. Hardy from 1918. We also obtain new results involving fractional derivatives of Canavati and Caputo types as ...

Let f be in the localized nonisotropic Sobolev space W 1,p loc (H ) on the n-dimensional Heisenberg group H = C × R, where 1 ≤ p < Q and Q = 2n + 2 is the homogeneous dimension of H. Suppose that the subelliptic gradient is gloablly L integrable, i.e., Hn |∇Hnf |pdu is finite. We prove a Poincaré inequality for f on the entire space H. Using this inequality we prove that the function f subtract...

This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints test H: h(1) > 0 versus K: 1 E RK. We extend these re...

In this article we investigate spectral properties of the coupling H +Vλ, where H = −iα · ∇ +mβ is the free Dirac operator in R, m > 0 and Vλ is an electrostatic shell potential (which depends on a parameter λ ∈ R) located on the boundary of a smooth domain in R. Our main result is an isoperimetric-type inequality for the admissible range of λ’s for which the coupling H + Vλ generates pure poin...

In this   paper,  we   present  some  refinements  of the   famous Young  type  inequality.   As  application  of   our   result, we  obtain  some  matrix inequalities   for   the  Hilbert-Schmidt norm  and   the  trace   norm. The results    obtained   in  this  paper  can  be   viewed   as  refinement  of  the   derived  results   by  H.  Kai  [Young  type  inequalities  for matrices,  J.  Ea...

Part 1: In this problem we will examine randomized classifiers. Define, for any point x, a probabilty distribution q(c|x) over the class labels c. We define a randomized decision rule given by h r (x; q) = c r , for c r ∼ q(c|x). The risk R(h r ; q) for this classifier is given by the expectation over all possible outcomes such that R(h r ; q) = x C c=1 C c =1 L(c , c)q(c r = c |x)p(x, y = c)dx...

Let H(S) be the Hardy space on the unit sphere S in C. We establish a local inequality for Hankel operators Hf = (1 − P )Mf |H(S). As an application of this local inequality, we characterize the membership of Hf in the Lorentz-like ideal C p , 2n < p <∞.