arXiv subjects

Tibor Kiss

Publications and source records attributed to Tibor Kiss.

15 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 P\'ales 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

Non-symmetrically $t$-affine functions revisited

In 2014, Michal Lewicki and Andrzej Olbry\'s proved that if a real valued function $f$ defined on the real line satisfies the conditional functional equation \[ f(tx + (1-t)y) = t f(x) + (1-t) f(y),\qquad x\leq y, \] called non-symmetrically $t$-affine, then it is $t$-affine. That is, they concluded that $f$ must fulfill the above equality without any restriction on $x$ and $y$. In the current study, first we show that the above conditional equation implies that the function in question is locally $t$-affine. Then we derive $t$-affinity on open intervals. Finally, we formulate our main result, which generalizes the theorem of Lewicki and Olbry\'s for any subinterval of $\mathbb{R}$.

math.CA↗

arXiv2026

Strictly nonexpansive, strictly monotone quasi graph-additive functions

In this paper, we provide a negative answer to an open problem concerning the functional equation \[f(f(-x)+x)=f(-f(x))+f(x),\] namely by showing that the family of continuous solutions is too rich to admit a complete description. Instead, we characterize the solutions within a certain subfamily.

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

Improved regularity for a composite functional equation stemming from the theory of means

In this paper we describe the solutions of the functional equation \begin{equation*} F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G \big(g_1(x)+g_2(y)) \end{equation*} defined on an open subinterval of $ \mathbb{R} $. Improving previous results we assume differentiability on each involved function, eliminate a former condition on $ g'_1 $ and $ g'_2$, moreover we determine a brand new family of solutions. We also present a particular member of this class as an example. In order to achieve this, we strengthen known results about certain auxiliary functional equations as well.

math.CA↗

arXiv2026

A Counterexample to Matkowski's Conjecture for Quasi Graph-Additive Functions

In this paper we investigate a conjecture of Janusz Matkowski concerning the continuous solutions of the functional equation \[ f\big(f(-x)+x\big)=f\big(-f(x)\big)+f(x),\qquad x\in\mathbb{R}. \] Matkowski conjectured that all continuous solutions must necessarily be linear on both the negative and the positive half-line. We show, however, that the family of continuous solutions to the equation in question is far richer than anticipated: there exist continuous solutions that admit an arbitrary part. In addition, we provide a sufficient condition which, in the continuous setting, enforces the conclusion predicted by Matkowski's Conjecture.

math.CA↗

arXiv2024

On the $σ$-balancing property of multivariate generalized quasi-arithmetic means

The aim of this paper is to characterize the so-called $σ$-balancing property in the class of generalized quasi-arithmetic means. In general, the question is whether those elements of a given family of means that possess this property are quasi-arithmetic. The first result in the latter direction is due to G. Aumann who showed that a balanced complex mean is necessariliy quasi-arithmetic provided that it is analytic. Then Aumann characterized quasi-arithmetic means among Cauchy means in terms of the balancing property. These results date back to the 1930s. In 2015, Lucio R. Berrone, generalizing balancedness, concluded that a mean having that more general property is quasi-arithmetic if it is symmetric, strict and continuously differentiable. A common feature of these results is that they assume a certain order of differentiability of the mean whether or not it is a natural condition. In 2020, the balancing property was characterized in the family of generalized quasi-arithmetic means of two variables under only natural conditions, namely continuity and strict monotonicity of their generating functions. Here we extend the corresponding result for multivariate generalized quasi-arithmetic means by relaxing the conditions on the generating functions and considering the more general $σ$-balancing property.

math.CA↗

arXiv2023

A Pexider equation containing the aritmetic mean

In this paper we determine the solutions $(φ,f_1,f_2)$ of the Pexider functional equation \[φ\Big(\frac{x+y}2\Big)\big(f_1(x)-f_2(y)\big)=0,\qquad (x,y)\in I_1\times I_2,\] where $I_1$ and $I_2$ are nonempty open subintervals. Special cases of the above equation regularly arise in problems with two-variable means. We show that, under a rather weak regularity condition, the coordinate-functions of a typical solution of the equation are constant over several subintervals of their domain. The regularity condition in question will be that the set of zeros of $φ$ is closed. We also discuss particular solutions where this condition is not met.

math.CA↗

arXiv2022

Regular solutions of a functional equation derived from the invariance problem of Matkowski means

The main result of the present paper is about the solutions of the functional equation \Eq{*}{ F\Big(\frac{x+y}2\Big)+f_1(x)+f_2(y)=G(g_1(x)+g_2(y)),\qquad x,y\in I, } derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where $F,f_1,f_2,g_1,g_2:I\to\mathbb{R}$ and $G:g_1(I)+g_2(I)\to\mathbb{R}$ are the unknown functions assumed to be continuously differentiable with $0\notin g'_1(I)\cup g'_2(I)$, and the set $I$ stands for a nonempty open subinterval of $\mathbb{R}$. In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that $F$ is affine on $I$ and $g_1$ and $g_2$ are continuous functions strictly monotone in the same sense, and secondly that $g_1$ and $g_2$ are invertible affine functions with a common additive part.

math.CA↗

arXiv2020

On the balancing property of Matkowski means

Let $I\subseteq\mathbb{R}$ be a nonempty open subinterval. We say that a two-variable mean $M:I\times I\to\mathbb{R}$ enjoys the \emph{balancing property} if, for all $x,y\in I$, the equality \begin{equation}\tag{1} M\big(M(x,M(x,y)),M(M(x,y),y)\big)=M(x,y) \end{equation} holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, that $M$ is \emph{analytic}, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes \emph{regular} quasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question are \emph{continuously differentiable}. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class of \emph{Matkowski means}.

math.CA↗

arXiv2018

On a functional equation related to two-variable Cauchy means

In this paper, we are dealing with the solution of the functional equation $$ φ\Big(\frac{x+y}2\Big)(f(x)-f(y))=F(x)-F(y), $$ concerning the unknown functions $φ,f$ and $F$ defined on a same open subinterval of the reals. Improving the previous results related to this topic, we describe the solution triplets $(φ,f,F)$ assuming only the continuity of $φ$. As an application, under natural conditions, we also solve the equality problem of two-variable Cauchy means and two-variable quasi-arithmetic means.

math.CA↗

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 $F,G,H,g,h$.

math.CA↗

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.

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.

math.CA↗

arXiv1996

Integrating Syntactic and Prosodic Information for the Efficient Detection of Empty Categories

We describe a number of experiments that demonstrate the usefulness of prosodic information for a processing module which parses spoken utterances with a feature-based grammar employing empty categories. We show that by requiring certain prosodic properties from those positions in the input where the presence of an empty category has to be hypothesized, a derivation can be accomplished more efficiently. The approach has been implemented in the machine translation project VERBMOBIL and results in a significant reduction of the work-load for the parser.

cmp-lg↗