In this paper a refinement of a particular case of an inequality of Radon, referred in the literature as Bergström's inequality, is presented. Later on, this result is used to obtain refinements of some known inequalities among them the classical inequality of Cauchy. Finally, two elementary numerical inequalities are also given. (2000): 26D15. Mathematics subject classification

The efficiency of algorithms for solving nonlinear equations is a measure of comparison between different iterative methods. In the case of scalar equations two parameters are considered as it is well-known, but frequently in recent literature inaccurate generalizations combining these parameters are used when solving systems of nonlinear equations. Our goal in this paper is to clarify the concept of the efficiency in the multi-dimensional case. To do it we present a detailed definition of the computational efficiency. The relation between the efficiency parameters in scalar and vectorial cases is analyzed in detail and tested in two numerical examples.

In this paper two families of cyclic inequalities in three variables are studied. More precisely, necessary and sufficient conditions in order that the cyclic sums ∑x m (x + y) n and ∑ s α (x)s β (x + y) are non-negative are stated and proven. Here m,n are positive integers, α,β are positive real numbers and s α (x) = x|x| α−1 , x ∈ R * ,s α (0) = 0. Mathematics subject classification (2010): 26D15.