107 research outputs found
Reducible means and reducible inequalities
It is well-known that if a real valued function acting on a convex set
satisfies the -variable Jensen inequality, for some natural number , then, for all , it fulfills the -variable Jensen
inequality as well. In other words, the arithmetic mean and the Jensen
inequality (as a convexity property) are both reducible. Motivated by this
phenomenon, we investigate this property concerning more general means and
convexity notions. We introduce a wide class of means which generalize the
well-known means for arbitrary linear spaces and enjoy a so-called reducibility
property. Finally, we give a sufficient condition for the reducibility of the
-convexity property of functions and also for H\"older--Minkowski type
inequalities
On Kedlaya type inequalities for weighted means
In 2016 we proved that for every symmetric, repetition invariant and Jensen
concave mean the Kedlaya-type inequality holds for an
arbitrary ( stands for the arithmetic mean). We are going
to prove the weighted counterpart of this inequality. More precisely, if
is a vector with corresponding (non-normalized) weights
and denotes the weighted mean then, under
analogous conditions on , the inequality holds for every and such that the sequence
is decreasing.Comment: J. Inequal. Appl. (2018
Abstract composition rule for relativistic kinetic energy in the thermodynamical limit
We demonstrate by simple mathematical considerations that a power-law tailed
distribution in the kinetic energy of relativistic particles can be a limiting
distribution seen in relativistic heavy ion experiments. We prove that the
infinite repetition of an arbitrary composition rule on an infinitesimal amount
leads to a rule with a formal logarithm. As a consequence the stationary
distribution of energy in the thermodynamical limit follows the composed
function of the Boltzmann-Gibbs exponential with this formal logarithm. In
particular, interactions described as solely functions of the relative
four-momentum squared lead to kinetic energy distributions of the
Tsallis-Pareto (cut power-law) form in the high energy limit.Comment: Submitted to Europhysics Letters. LaTeX, 3 eps figure
On the invariance equation for two-variable weighted nonsymmetric Bajraktarevi\'c means
The purpose of this paper is to investigate the invariance of the arithmetic
mean with respect to two weighted Bajraktarevi\'c means, i.e., to solve the
functional equation where are unknown continuous
functions such that are nowhere zero on , the ratio functions ,
are strictly monotone on , and are constants
different from each other. By the main result of this paper, the solutions of
the above invariance equation can be expressed either in terms of hyperbolic
functions or in terms of trigonometric functions and an additional weight
function. For the necessity part of this result, we will assume that
are four times continuously differentiable
Cantor type functions in non-integer bases
Cantor's ternary function is generalized to arbitrary base-change functions
in non-integer bases. Some of them share the curious properties of Cantor's
function, while others behave quite differently
A composite functional equation from algebraic aspect
In this paper we discuss the composite functional equation
f(x+2f(y))=f(x)+y+f(y)
on an Abelian group. This equation originates from Problem 10854 of the American Mathematical Monthly. We give an algebraic description of the solutions on uniquely 3-divisible Abelian groups, and then we construct all solutions f of this equation on finite Abelian groups without elements of order 3 and on divisible Abelian groups without elements of order 3 including the additive group of real numbers
Some inequalities on generalized entropies
We give several inequalities on generalized entropies involving Tsallis
entropies, using some inequalities obtained by improvements of Young's
inequality. We also give a generalized Han's inequality.Comment: 15 page
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