Unified treatment of fractional integral inequalities via linear functionals

In the paper we prove several inequalities involving two isotonic linear functionals. We consider inequalities for functions with variable bounds, for Lipschitz and H\" older type functions etc. These results give us an elegant method for obtaining a number of inequalities for various kinds of fractional integral operators such as for the Riemann-Liouville fractional integral operator, the Hadamard fractional integral operator, fractional hyperqeometric integral and corresponding q-integrals

Universality of free homogeneous sums in every dimension

We prove a general multidimensional invariance principle for a family of U-statistics based on freely independent non-commutative random variables of the type $U_n(S)$, where $U_n(x)$ is the $n$-th Chebyshev polynomial and $S$ is a standard semicircular element on a fixed $W^{\ast}$-probability space. As a consequence, we deduce that homogeneous sums based on random variables of this type are universal with respect to both semicircular and free Poisson approximations. Our results are stated in a general multidimensional setting and can be seen as a genuine extension of some recent findings by Deya and Nourdin; our techniques are based on the combination of the free Lindeberg method and the Fourth moment Theorem

Functional inequalities for the Bickley function

In this paper our aim is to deduce some complete monotonicity properties and functional inequalities for the Bickley function. The key tools in our proofs are the classical integral inequalities, like Chebyshev, H\"older-Rogers, Cauchy-Schwarz, Carlson and Gr\"uss inequalities, as well as the monotone form of l'Hospital's rule. Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Bickley function.Comment: 10 page

Stability analysis of spectral methods for hyperbolic initial-boundary value systems

A constant coefficient hyperbolic system in one space variable, with zero initial data is discussed. Dissipative boundary conditions are imposed at the two points x = + or - 1. This problem is discretized by a spectral approximation in space. Sufficient conditions under which the spectral numerical solution is stable are demonstrated - moreover, these conditions have to be checked only for scalar equations. The stability theorems take the form of explicit bounds for the norm of the solution in terms of the boundary data. The dependence of these bounds on N, the number of points in the domain (or equivalently the degree of the polynomials involved), is investigated for a class of standard spectral methods, including Chebyshev and Legendre collocations