arXiv subjects

Paweł Pasteczka

Publications and source records attributed to Paweł Pasteczka.

At least 19 recordsLinked to original sources

arXiv2026

Equality problem for generalized quasiarithmetic means generated by discontinuous strictly monotonic functions

We study the equality problem of generalized quasiarithmetic means for a strictly monotonic generator $f$ that is not necessarily continuous. We provide two sufficient conditions that lead to a conclusion analogous to the result of Páles and Pasteczka. We show through an example that in our case, without any extra conditions, the generator functions cannot be expected to be affine transformations of each other over the whole domain. In the remaining case, we consider the appropriate inverses of the functions, which implies that the scaling factor must coincide across the various regions of continuity.

math.CA↗

arXiv2026

Lattice-like property of quasi-arithmetic means: revisited

We show that every family of quasi-arithmetic means generated by (a subset of) $\mathcal{C}^1$ functions with nonvanishing derivative which is bounded (from below or from above) by a quasi-arithmetic mean, possesses the best (lower or upper) bound which is a quasi-arithmetic mean generated by a function belonging to the same family.

math.GM↗

arXiv2026

On subadditive quasi-arithmetic means

Let $f\colon \mathbb{R}_+\to\mathbb{R}$ be a continuous and strictly monotone function. In the main result of this paper, we show that, for a fixed $n\geq 2$, the $n$-variable mean $\mathscr{A}_f \colon \mathbb{R}_+^n \to \mathbb{R}_+$ defined by $$ \mathscr{A}_f(x_1,\dots,x_n):=f^{-1} \bigg( \frac{f(x_1)+\cdots+f(x_n)}n \bigg) $$ is subadditive if and only if $f$ is differentiable with a continuously semi-differentiable and nonvanishing first derivative, and there exists an $α\in[0,\infty]$ such that $f''_+:=(f')'_+$ is positive on $(0,α)$ and $f''_+=0$ on $[α,\infty)$, furthermore, $\frac{f'}{f''_+}$ is increasing and superadditive on $(0,α)$.

math.CA↗

arXiv2025

On the new smoothness class of means and its impact to mean-type mappings

We define so-called residual means, which have a Taylor expansion of the form $M(x)=\bar x +\tfrac12 ξ_M(\bar x) \text{Var}(x)+o(\|x-\bar x\|^α)$ for some $α>2$ and a single-variable function $ξ_M$ ($\bar x$ stands for the arithmetic mean of the vector $x$), and show that all symmetric means which are three times continuously differentiable are residual. We also calculate the value of residuum for quasideviation means and a few subclasses of this family. Later, we apply it to establish the limit of the sequence $\big(\frac{\text{Var}\ {\bf M}^{n+1}(x)}{(\text{Var}\ {\bf M}^n(x))^2}\big)_{n=1}^\infty$, where ${\bf M} \colon I^p\to I^p$ is a mean-type mapping consisting of $p$-variable residual means on an interval $I$, and $x \in I^p$ is a nonconstant vector.

math.CA↗

arXiv2025

Hölder- and Minkowski-type inequalities for generalized quasi-arithmetic means

The purpose of this paper is to establish several necessary and sufficient conditions to ensure the validity of a general functional inequality in terms of generalized quasi-arithmetic means. In particular cases, we consider Hölder-, Minkowski-, and Jensen-type inequalities. Generalized quasi-arithmetic means are defined by taking strictly monotone generating functions instead of strictly monotone and continuous ones.

math.CA↗

arXiv2024

Equality and comparison of generalized quasiarithmetic means

The purpose of this paper is to extend the definition of quasiarithmetic means by taking a strictly monotone generating function instead of a strictly monotone and continuous one. We establish the properties of such means and compare them to the analogous properties of standard quasiarithmetic means. The comparability and equality problems of generalized quasiarithmetic are also solved. We also provide an example of a mean which, depending on the underlying interval or on the number of variables, could be or could not be represented as a generalized quasiarithmetic mean.

math.CA↗

arXiv2024

Mathematical model of information bubbles on networks

The main goal of this paper to introduce a new model of evolvement of narratives (common opinions, information bubble) on networks. Our main tools come from invariant mean theory and graph theory. The case, when the root set of the network (influencers, news agencies, etc.) is ergodic is fully discussed. The other possibility, when the root contains more than one component is partially discussed and it could be a motivation for further research.

cs.SI↗

arXiv2024

Multivariable generalizations of bivariate means via invariance

For a given $p$-variable mean $M \colon I^p \to I$ ($I$ is a subinterval of $\mathbb{R}$), following (Horwitz, 2002) and (Lawson and Lim, 2008), we can define (under certain assumption) its $(p+1)$-variable $β$-invariant extension as the unique solution $K \colon I^{p+1} \to I$ of the functional equation \begin{align*} K\big(M(x_2,\dots,x_{p+1})&,M(x_1,x_3,\dots,x_{p+1}),\dots,M(x_1,\dots,x_p)\big)\\ &=K(x_1,\dots,x_{p+1}), \text{ for all }x_1,\dots,x_{p+1} \in I \end{align*} in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.

math.DS↗

arXiv2024

Pexider invariance equation for embeddable mean-type mappings

We prove that whenever $M_1,\dots,M_n\colon I^k \to I$, ($n,k \in \mathbb{N}$) are symmetric, continuous means on the interval $I$ and $S_1,\dots,S_m\colon I^k \to I$ ($m <n$) satisfies a sort of embeddability assumptions then for every continuous function $μ\colon I^n \to \mathbb{R}$ which is strictly monotone in each coordinate, the functional equation $$ μ(S_1(v),\dots,S_m(v),\underbrace{F(v),\dots,F(v)}_{(n-m)\text{ times}})=μ(M_1(v),\dots,M_n(v)) $$ has the unique solution $F=F_μ\colon I^k \to I$ which is a mean. We deliver some sufficient conditions so that $F_μ$ is well-defined (in particular uniquely determined) and study its properties. The background of this research is to provide a broad overview of the family of Beta-type means introduced in (Himmel and Matkowski, 2018).

math.CA↗

arXiv2023

Extension theorem for simultaneous q-difference equations and some its consequences

Given a set $T \subset (0, +\infty)$, intervals $I\subset (0, +\infty)$ and $J\subset {\mathbb R}$, as well as functions $g_t:I\times J\rightarrow J$ with $t$'s running through the set \[ T^{\ast}:=T \cup \big\{t^{-1}\colon t \in T\big\}\cup\{1\} \] we study the simultaneous $q$-difference equations \[ φ(tx)=g_t\left(x,φ(x)\right), \qquad t \in T^{\ast}, \] postulated for $x \in I\cap t^{-1}I$; here the unknown function $φ$ is assumed to map $I$ into $J$. We prove an Extension theorem stating that if $φ$ is continuous [analytic] on a nontrivial subinterval of $I$, then $φ$ is continuous [analytic] provided $g_t, t \in T^{\ast}$, are continuous [analytic]. The crucial assumption of the Extension theorem is formulated with the help of the so-called limit ratio $R_T$ which is a uniquely determined number from $[1,+\infty]$, characterising some density property of the set $T^{\ast}$. As an application of the Extension theorem we find the form of all continuous on a subinterval of $I$ solutions $φ:I \rightarrow {\mathbb R}$ of the simultaneous equations \[ φ(tx)=φ(x)+c(t)x^p, \qquad t\in T, \] where $c:T \rightarrow {\mathbb R}$ is an arbitrary function, $p$ is a given real number and $\sup I > R_T \inf I$.

math.CA↗

arXiv2023

On the invariance equation for means of generalized power growth

We generalize the result of (Witkowski, 2014) which binds orders of homogeneous, symmetric means $M,N,K \colon\mathbb{R}_+^2 \to \mathbb{R}_+$ of power growth that satisfy the invariance equation $K(M(x,y),N(x,y))=K(x,y)$ to the broader class of means. Moreover, we define the lower- and the upper-order which gives us insight into the order of the solution of this equation in the case when means do not belong to this class.

math.CA↗

arXiv2023

Estimating the Hardy constant of nonconcave homogenus quasideviation means

In this paper, we consider homogeneous quasideviation means generated by real functions (defined on $(0,\infty)$) which are concave around the point $1$ and possess certain upper estimates near $0$ and $\infty$. It turns out that their concave envelopes can be completely determined. Using this description, we establish sufficient conditions for the Hardy property of the homogeneous quasideviation mean and we also furnish an upper estimates for its Hardy constant.

math.CA↗

arXiv2023

Invariance property for extended means

e study the properties of the mean-type mappings ${\bf M}\colon I^p \to I^p$ of the form $${\bf M}(x_1,\dots,x_p):=\big(M_1(x_{α_{1,1}},\dots,x_{α_{1,d_1}}),\dots,M_p(x_{α_{p,1}},\dots,x_{α_{p,d_p}})\big),$$ where $p$ and $d_i$-s are positive integers, each $M_i$ is a $d_i$-variable mean on an interval $I \subset \mathbb{R}$, and $α_{i,j}$-s are elements from $\{1,\dots,p\}$. We show that, under some natural assumption on $M_i$-s, the problem of existing the unique $\bf M$-invariant mean can be reduced to the ergodicity of the directed graph with vertexes $\{1,\dots,p\}$ and edges $\{(α_{i,j},i) \colon i,j \text{ admissible}\}$.

math.CA↗

arXiv2023

On the Jensen convexity of quasideviation and Bajraktarević means

Motivated by the characterization theorem about the Jensen convexity of quasiarithmetic means obtained by the authors in 2021, our main goal is to establish a characterization of the Jensen convexity of quasideviation as well as of Bajraktarević means without any additional and unnatural regularity assumptions.

math.CA↗

arXiv2022

On properties of weighted Hardy constant for means

For a given weighted mean $\mathscr{M}$ defined on a subinterval of $\mathbb{R}_+$ and a sequence of weights $λ=(λ_n)_{n=1}^\infty$ we define a Hardy constant $\mathscr H(λ)$ as the smallest extended real number such that $$ \sum_{n=1}^\infty λ_n \mathscr{M}\big((x_1,\dots,x_n),(λ_1,\dots,λ_n)\big) \le \mathscr H(λ) \cdot \sum_{n=1}^\infty λ_n x_n \text{ for all }x \in \ell^1(λ).$$ The aim of this note is to present a comprehensive study of the mapping $\mathscr H$. For example we prove that it is lower semicontinuous in the pointwise topology. Moreover we show that whenever $\mathscr{M}$ is a monotone and Jensen-concave mean which is continuous in its weights then $\mathscr H$ is monotone with respect to the partitioning of the vector. Finally we deliver some sufficient conditions for $λ$ to validate the equality $\mathscr H(λ)=\sup \mathscr H$ for every symmetric and monotone mean.

math.CA↗

arXiv2021

On negative results concerning weak-Hardy means

We establish the test which allows to show that a mean does not admit a weak-Hardy property. As a result we prove that Hardy and weak-Hardy properties are equivalent in the class of homogeneous, symmetric, repetition invariant, and Jensen concave mean on $\mathbb{R}_+$. More precisely, for every mean $\mathscr{M} \colon \bigcup_{n=1}^\infty \mathbb{R}_+^n \to \mathbb{R}$ as above, the inequality $$\mathscr{M}(a_1)+\mathscr{M}(a_1,a_2)+\dots<\infty$$ holds for all $a \in \ell^1(\mathbb{R}_+)$ if and only if there exists a positive, real constant $C$ (depending only on $\mathscr{M}$) such that $$\mathscr{M}(a_1)+\mathscr{M}(a_1,a_2)+\dots<C \cdot (a_1+a_2+\cdots)$$ for every sequence $a \in \ell^1(\mathbb{R}_+)$.

math.CA↗

arXiv2021

Estimating the Hardy constant of nonconcave Gini means

The extension of the Hardy-Knopp-Carleman inequality to several classes of means was the subject of numerous papers. In the class of Gini means the Hardy property was characterized in 2015 by the second author. The precise value of the associated Hardy constant was only established for concave Gini means by the authors in 2016. The determination Hardy constant for nonconcave Gini means is still an open problem. The main goal of this paper is to establish sharper upper bounds for the Hardy constant in this case. The method is to construct a homogeneous and concave quasideviation mean which majorizes the nonconcave Gini mean and for which the Hardy constant can be computed.

math.CA↗

arXiv2021

There is at most one continuous invariant mean

We show that, for a (not necessarily continuous) weakly contractive mean-type mapping $\mathbf{M} \colon I^p\to I^p$ (where $I$ is an interval and $p \in \mathbb{N}$), the functional equation $K \circ \mathbf{M}=K$ has at most one solution in the family of continuous means $K \colon I^p \to I$. Some general approach to the latter equation is also given.

math.CA↗