On the local and global comparison of generalized Bajraktarevi\'c 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 $\mu$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;\mu}:I^d\to I$ is defined by $$M_{f,g,m;\mu}(\pmb{x}) :=\left(\frac{f}{g}\right)^{-1}\left( \frac{\int_T f\big(m(x_1,\dots,x_d,t)\big) d\mu(t)} {\int_T g\big(m(x_1,\dots,x_d,t)\big) d\mu(t)}\right) \qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d).$$ 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 $\mu,\nu$ such that the comparison inequality $$M_{f,g,m;\mu}(\pmb{x})\leq M_{h,k,n;\nu}(\pmb{x}) \qquad(\pmb{x}\in I^d)$$ be satisfied

Függvényegyenletek és egyenlőtlenségek = Functional equations and inequalities

A kutatásban 19 fő vett részt. A kutatócsoport tagjai 2003 és 2006 között 3 könyvet, 103 referált nemzetközi folyóiratban, vagy konferenciakiadványban megjelent tudományos dolgozatot, további 15 közlésre elfogadott dolgozatot, 6 PhD, 1 habilitációs és 2 MTA doktori értekezést készítettek, továbbá 208 tudományos előadást tartottak. A nem-iteratív függvényegyenletek regularitáselméletének monografikus összefoglalását adja Járai Antal 2005-ben a Kluwer kiadónál megjelent könyve. Ez mű az elmúlt két évtizedben a kutatócsoport által elért eredményeket egységes szemléletmódban tárgyalja és új alkalmazásokat is bemutat. Az iteratív függvényegyenletek és az un. invariancia-egyenlet regularitási vizsgálataiban is számos új módszer és eredmény született. A spektrálszintézis és spektrálanalízis kutatásában döntő áttörést hoztak Székelyhidi László kutatásai. Kiderült, hogy bizonyos diszkrét Abel-csoportokon a spektrálszintézis nem teljesülhet, ezzel egy több évtizede fennálló sejtés is megdőlt. A témakör elméletének alapvető és lezáró eredményeit tartalmazó monográfia 2006-ban a Springer kiadónál jelent meg. A lineáris operátorok és függvényterek megőrzési problémáinak összefoglalását adja Molnár Lajosnak a Springer Lecture Notes sorozatában 2007-ben megjelent könyve. A Hadamard-egyenlőtlenség magasabbrendű általánosításainak és a közelítőleg konvex függvényeknek a vizsgálata is számos új fejleményet hozott. | The research was carried out by 19 researchers. The members of the research team published 3 books, 103 research papers in refereed journals or conference proceedings, 15 papers accepted for publication, furthermore, 6 PhD, 1 habilitation and 2 DSc dissertations in the period 2003-2006. The monographic summary of the regularity theory of non-iterative functional equations was published by Antal Járai at the Kluwer Academic Publishers. This work treats the results of the research team obtained in the last two decades in a unified manner and also presents new applications. The investigation of the regularity problems of the iterative functional equations and the so-called invariance equation lead to a number of new methods and results. The investigation of spectral synthesis and spectral analysis reached a breakthrough point due to László Székelyhidi's results. It turned out that the spectral synthesi fails to hold on certain discrete Abelian groups. This result negatively answered a longstanding conjecture. The basic and key results of the subject were summarized in a book published by Springer Verlag. A book by Lajos Molnár, published in the Lecture Notes of Springer in 2007, summarizes the results obtained on preserver problems of linear operators and functions spaces. There were also new and interesting developments in the investigation of higher-order generalizations of the Hadamard inequality and approximately convex functions

Aggregate Bound Choices about Random and Nonrandom Goods Studied via a Nonlinear Analysis

In this paper, bound choices are made after summarizing a finite number of alternatives. This means that each choice is always the barycenter of masses distributed over a finite set of alternatives. More than two marginal goods at a time are not handled. This is because a quadratic metric is used. In our models, two marginal goods give rise to a joint good, so aggregate bound choices are shown. The variability of choice for two marginal goods that are the components of a multiple good is studied. The weak axiom of revealed preference is checked and mean quadratic differences connected with multiple goods are proposed. In this paper, many differences from vast majority of current research about choices and preferences appear. First of all, conditions of certainty are viewed to be as an extreme simplification. In fact, in almost all circumstances, and at all times, we all find ourselves in a state of uncertainty. Secondly, the two notions, probability and utility, on which the correct criterion of decision-making depends, are treated inside linear spaces over R having a different dimension in accordance with the pure subjectivistic point of vie

Tensors Associated with Mean Quadratic Differences Explaining the Riskiness of Portfolios of Financial Assets

Bound choices such as portfolio choices are studied in an aggregate fashion using an extension of the notion of barycenter of masses. This paper answers the question of whether such an extension is a natural fashion of studying bound choices or not. Given n risky assets, the question of why it is appropriate to treat only two risky assets at a time inside the budget set of the decision-maker is handled in this paper. Two risky assets are two goods. They are two marginal goods. The question of why they always give rise to a joint good inside the budget set of the decision-maker is addressed by this research work. A single risky asset is viewed as a double one using four nonparametric joint distributions of probability. The variability of a joint distribution of probability always depends on the state of information and knowledge associated with a given decision-maker. For this reason, two variability tensors are defined to identify the riskiness of the same risky asset. A multilinear version of the Sharpe ratio is shown. It is based on tensors. After computing the expected return on an n-risky asset portfolio, its riskiness is obtained using mean quadratic differences developed through tensor