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Numerical radius inequalities for finite sums of operators

Mirmostafaee A. K., Rahpeyma O. P., Omidvar M. E.

Demonstratio Mathematica

2014

Vol. 47, nr 4

963--970

In this paper, we obtain some sharp inequalities for numerical radius of finite sums of operators. Moreover, we give some applications of our result in estimation of spectral radius. We also compare our results with some known results.

Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels

Castro L. P., Saitoh S., Tuan N. M.

Opuscula Mathematica

2012

Vol. 32, no. 4

633-646

This paper introduces a general concept of convolutions by means of the theory of reproducing kernels which turns out to be useful for several concrete examples and applications. Consequent properties are exposed (including, in particular, associated norm inequalities).

Maximal inequalities for stochastic integrals

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

Vol. 58, no 3

273-287

We find the optimal universal constant Cp (1 < p ≤ ∞) in the following inequality. If X = (Xt)t>o is a martingale and Y = [wzór] for some predictable process H taking values in [-1,1], then E[wzór].

Sharp norm inequalities for stochastic integrals in which the integrator is a nonnegative supermartingale

Osękowski A.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 1

29--42

The paper is devoted to sharp inequalities between moments of nonnegative supermartingales and their strong subordinates. Analogous estimates hold true for stochastic integrals with respect to a nonnegative right-continuous supermartingale. Similar inequalities are established for smooth functions on Euclidean domains.

Sharp norm inequalities for martingales and their differential subordinates

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 4

373-385

Suppose ƒ = (ƒn), g = (gn) are martingales with respect to the same filtration, satisfying |ƒn-ƒn-i| ≤ |gn -gn-1|, n = 1,2,..., with probability 1. Under some assumptions on ƒo, go and an additional condition that one of the processes is nonnegative, some sharp inequalities between the pth norms of ƒ and g, 0 < p < ∞, are established. As an application, related sharp inequalities for stochastic integrals and harmonic functions are obtained.