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Zsolt Páles

Publications and source records attributed to Zsolt Páles.

At least 73 records · Page 4Linked to original sources
arXiv2018

On Kedlaya type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$ \mathscr{A}\big(x_1,\mathscr{M}(x_1,x_2),\ldots,\mathscr{M}(x_1,\ldots,x_n)\big)\le \mathscr{M} \big(x_1, \mathscr{A}(x_1,x_2),\ldots,\mathscr{A}(x_1,\ldots,x_n)\big) $$ holds for an arbitrary $(x_n)$ ($\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_n)$ is a vector with corresponding (non-normalized) weights $(λ_n)$ and $\mathscr{M}_{i=1}^n(x_i,λ_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$ \mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,λ_j),\:λ_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,λ_j),\:λ_i\big) $$ holds for every $(x_n)$ and $(λ_n)$ such that the sequence $(\frac{λ_k}{λ_1+\cdots+λ_k})$ is decreasing.

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arXiv2018

A new characterization of convexity with respect to Chebyshev systems

The notion of $n$th order convexity in the sense of Hopf and Popoviciu is defined via the nonnegativity of the $(n+1)$st order divided differences of a given real-valued function. In view of the well-known recursive formula for divided differences, the nonnegativity of $(n+1)$st order divided differences is equivalent to the $(n-k-1)$st order convexity of the $k$th order divided differences which provides a characterization of $n$th order convexity. The aim of this paper is to apply the notion of higher-order divided differences in the context of convexity with respect to Chebyshev systems introduced by Karlin in 1968. Using a determinant identity of Sylvester, we then establish a formula for the generalized divided differences which enables us to obtain a new characterization of convexity with respect to Chebyshev systems. Our result generalizes that of Wąsowicz which was obtained in 2006. As an application, we derive a necessary condition for functions which can be written as the difference of two functions convex with respect to a given Chebyshev system.

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arXiv2018

On the best Hardy constant for quasi-arithmetic means and homogeneous deviation means

The aim of this paper is to characterize the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n $$ for all positive sequences $(x_n)$ with some finite positive constant $C$. The smallest constant $C$ satisfying this property is called the Hardy constant of the mean $M$. In this paper we determine the Hardy constant in the cases when the mean $M$ is either a concave quasi-arithmetic or a concave and homogeneous deviation mean.

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arXiv2018

On the invariance equation for two-variable weighted nonsymmetric Bajraktarević means

The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarević means, i.e., to solve the functional equation $$ \bigg(\frac{f}{g}\bigg)^{\!\!-1}\!\!\bigg(\frac{tf(x)+sf(y)}{tg(x)+sg(y)}\bigg) +\bigg(\frac{h}{k}\bigg)^{\!\!-1}\!\!\bigg(\frac{sh(x)+th(y)}{sk(x)+tk(y)}\bigg)=x+y \qquad(x,y\in I), $$ where $f,g,h,k:I\to\mathbb{R}$ are unknown continuous functions such that $g,k$ are nowhere zero on $I$, the ratio functions $f/g$, $h/k$ are strictly monotone on $I$, and $t,s\in\mathbb{R}_+$ 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 $f,g,h,k:I\to\mathbb{R}$ are four times continuously differentiable.

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arXiv2017

On derivations with respect to finite sets of smooth functions

The purpose of this paper is to show that functions that derivate the two-variable product function and one of the exponential, trigonometric or hyperbolic functions are also standard derivations. The more general problem considered is to describe finite sets of differentiable functions such that derivations with respect to this set are automatically standard derivations.

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arXiv2017

On Hardy type inequalities for weighted means

The aim of this paper is to establish weighted Hardy type inequality in a broad family of means. In other words, for a fixed vector of weights $(λ_n)_{n=1}^\infty$ and a weighted mean $\mathscr{M}$, we search for the smallest number $C$ such that $$\sum_{n=1}^{\infty} λ_n \mathscr{M} \big((x_1,\dots,x_n),(λ_1,\dots,λ_n)\big) \le C \sum_{n=1}^{\infty} λ_nx_n \text{ for all admissible }x.$$ The main results provide a definite answer in the case when $\mathcal{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, if $\mathcal{M}$ is symmetric, Jensen-concave, and the sequence $\big(\tfrac{λ_n}{λ_1+\cdots+λ_n}\big)$ is nonincreasing. In addition, it is proved that if $\mathcal{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $λ$ is the constant vector.

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arXiv2017

Equality and homogeneity of generalized integral means

Given two continuous functions $f,g:I\to\mathbb{R}$ such that $g$ is positive and $f/g$ is strictly monotone, a measurable space $(T,A)$, a measurable family of $d$-variable means $m: I^d\times T\to I$, and a probability measure $μ$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;μ}:I^d\to I$ is defined by $$ M_{f,g,m;μ}(\pmb{x}) :=\left(\frac{f}{g}\right)^{-1}\left( \frac{\int_T f\big(m(x_1,\dots,x_d,t)\big)\,dμ(t)} {\int_T g\big(m(x_1,\dots,x_d,t)\big)\,dμ(t)}\right) \qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d). $$ The aim of this paper is to solve the equality and homogeneity problems of these means, i.e., to find conditions for the generating functions $(f,g)$ and $(h,k)$, for the family of means $m$, and for the measure $μ$ such that the equality $$ M_{f,g,m;μ}(\pmb{x})=M_{h,k,m;μ}(\pmb{x}) \qquad(\pmb{x}\in I^d) $$ and the homogeneity property $$ M_{f,g,m;μ}(λ\pmb{x})=λM_{f,g,m;μ}(\pmb{x}) \qquad(λ>0,\,\pmb{x},λ\pmb{x}\in I^d), $$ respectively, be satisfied.

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arXiv2017

Support theorems in abstract settings

In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain space, we allow algebraic structures equipped with families of algebraic operations whose operations are mutually distributive with respect to each other. We introduce several new concepts in such algebraic structures, the notions of convex set, extreme set, and interior point with respect to a given family of operations, furthermore, we describe their most basic and required properties. In the context of the range space, we introduce the notion of completeness of a partially ordered set with respect to the existence of the infimum of lower bounded chains, we also offer several sufficient condition which imply this property. For instance, the order generated by a sharp cone in a vector space turns out to possess this completeness property. By taking several particular cases, we deduce support and extension theorems in various classical and important settings.

math.FA↗
arXiv2017

On a functional equation related to two-variable weighted quasi-arithmetic means

In this paper, we are going to describe the solutions of the functional equation $$ φ\Big(\frac{x+y}{2}\Big)(f(x)+f(y))=φ(x)f(x)+φ(y)f(y) $$ concerning the unknown functions $φ$ and $f$ defined on an open interval. In our main result only the continuity of the function $φ$ and a regularity property of the set of zeroes of $f$ are assumed. As application, we determine the solutions of the functional equation $$ G(g(u)-g(v))=H(h(u)+h(v))+F(u)+F(v) $$ under monotonicity and differentiability conditions on the unknown functions $F,G,H,g,h$.

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arXiv2017

On the local and global comparison of generalized Bajraktarević means

Given two continuous functions $f,g:I\to\mathbb{R}$ such that $g$ is positive and $f/g$ is strictly monotone, a measurable space $(T,A)$, a measurable family of $d$-variable means $m: I^d\times T\to I$, and a probability measure $μ$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;μ}:I^d\to I$ is defined by $$ M_{f,g,m;μ}(\pmb{x}) :=\left(\frac{f}{g}\right)^{-1}\left( \frac{\int_T f\big(m(x_1,\dots,x_d,t)\big) dμ(t)} {\int_T g\big(m(x_1,\dots,x_d,t)\big) dμ(t)}\right) \qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d). $$ The aim of this paper is to study the local and global comparison problem of these means, i.e., to find conditions for the generating functions $(f,g)$ and $(h,k)$, for the families of means $m$ and $n$, and for the measures $μ,ν$ such that the comparison inequality $$ M_{f,g,m;μ}(\pmb{x})\leq M_{h,k,n;ν}(\pmb{x}) \qquad(\pmb{x}\in I^d) $$ be satisfied.

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arXiv2016

Reducible means and reducible inequalities

It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-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 $(M,N)$-convexity property of functions and also for Hölder--Minkowski type inequalities.

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arXiv2016

Asymptotic stability of the Cauchy and Jensen functional equations

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations.

math.CA↗
arXiv2015

Convexity with respect to families of means

In this paper we investigate continuity properties of functions $f:\mathbb{R}_+\to\mathbb{R}_+$ that satisfy the $(p,q)$-Jensen convexity inequality $$ f\big(H_p(x,y)\big)\leq H_q(f(x),f(y)) \qquad(x,y>0), $$ where $H_p$ stands for the $p$th power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are $(p,p)$-Jensen convex for all positive rational number $p$. A counterpart of this result states that if $f$ is $(p,p)$-Jensen convex for all $p\in P\subseteq\mathbb{R}_+$, where $P$ is a set of positive Lebesgue measure, then $f$ must be continuous.

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arXiv2015

Convexity and a Stone-type theorem for convex sets in abelian semigroup setting

In this paper, two parallel notions of convexity of sets are introduced in the abelian semigroup setting. The connection of these notions to algebraic and to set-theoretic operations is investigated. A formula for the computation of the convex hull is derived. Finally, a Stone-type separation theorem for disjoint convex sets is established.

math.CA↗
arXiv2015

Characterization of the Hardy property of means and the best Hardy constants

The aim of this paper is to characterize in broad classes of means the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty \mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$ \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n $$ for all positive sequences $(x_n)$ with some finite positive constant $C$. One of the main results offers a characterization of Hardy means in the class of symmetric, increasing, Jensen concave and repetition invariant means and also a formula for the best constant $C$ satisfying the above inequality.

math.CA↗
arXiv2015

Implications between generalized convexity properties of real functions

Motivated by the well-known implications among $t$-convexity properties of real functions, analogous relations among the upper and lower $M$-convexity properties of real functions are established. More precisely, having an $n$-tuple $(M_1,\dots,M_n)$ of continuous two-variable means, the notion of the descendant of these means (which is also an $n$-tuple $(N_1,\dots,N_n)$ of two-variable means) is introduced. In particular, when all the means $M_i$ are weighted arithmetic, then the components of their descendants are also weighted arithmetic means. More general statements are obtained in terms of the generalized quasi-arithmetic or Matkowski means. The main results then state that if a function $f$ is $M_i$-convex for all $i\in\{1,\dots,n\}$, then it is also $N_i$-convex for all $i\in\{1,\dots,n\}$. Several consequences are discussed.

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arXiv2015

A General Mean Value Theorem

In this note a general a Cauchy-type mean value theorem for the ratio of functional determinants is offered. It generalizes Cauchy's and Taylor's mean value theorems as well as other classical mean value theorems.

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