57 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
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
Ulam type stability problems for alternative homomorphisms
We introduce an alternative homomorphism with respect to binary operations and investigate the Ulam type stability problem for such a mapping. The obtained results apply to Ulam type stability problems for several important functional equations.ArticleJOURNAL OF INEQUALITIES AND APPLICATIONS. 2014:228 (2014)journal articl
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
First and second order optimality conditions for optimal control problems of state constrained integral equations
This paper deals with optimal control problems of integral equations, with
initial-final and running state constraints. The order of a running state
constraint is defined in the setting of integral dynamics, and we work here
with constraints of arbitrary high orders. First and second-order necessary
conditions of optimality are obtained, as well as second-order sufficient
conditions
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