Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian rando...

We use Salem’s method [13, 14] to prove an inequality of Kwapień and Pełczyński concerning a lower bound for partial sums of series of bi-orthogonal vectors in a Hilbert space, or the dual vectors. This is applied to some lower bounds on L norms for orthogonal expansions.

Some sharp inequalities of Grüss type for sequences of vectors in real or complex inner product spaces are obtained. Applications for Jensen's inequality for convex functions defined on such spaces are also provided.

In this work, we deal with a Slater type inequality designed for symmetric convex function and collection of vectors transformed by doubly stochastic matrix. doing so, use an additional control function. the case when composition underlying is Schur-concave, such approach leads to refinement standard inequality. Special cases are also considered.

A new determinant inequality of positive semidefinite matrices is discovered and proved by us. This new inequality is useful for attacking and solving a variety of optimization problems arising from the design of wireless communication systems. I. A NEW DETERMINANT INEQUALITY The following notations are used throughout this article. The notations [·] and [·] stand for transpose and Hermitian tr...

In this article, we derive a new generalization of Chebyshev inequality for random vectors. We demonstrate that the new generalization is much less conservative than the classical generalization. 1 Classical Generalization of Chebyshev inequality The Chebyshev inequality discloses the fundamental relationship between the mean and variance of a random variable. Extensive research works have been...

We obtain some inequalities related to the powers of numerical radius inequalities of Hilbert space operators. Some results that employ the Hermite-Hadamard inequality for vectors in normed linear spaces are also obtained. We improve and generalize some inequalities with respect to Specht's ratio. Among them, we show that, if $A, Bin mathcal{B(mathcal{H})}$ satisfy in some conditions, it follow...

We investigate the universal linear inequalities that hold for the von Neumann entropies in a multi-party system, prepared in a stabiliser state. We demonstrate here that entropy vectors for stabiliser states satisfy, in addition to the classic inequalities, a type of linear rank inequalities associated with the combinatorial structure of normal subgroups of certain matrix groups. In the 4-part...