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 H\"older--Minkowski type inequalities

Generalized Cauchy means

Given two means M and N, the operator MM,NMM,N assigning to a given mean μ the mean MM,N(μ)(x,y)=M(μ(x,N(x,y)),μ(N(x,y),y)) was defined in Berrone and Moro (Aequationes Math 60:1–14, 2000) in connection with Cauchy means: the Cauchy mean generated by the pair f, g of continuous and strictly monotonic functions is the unique solution μ to the fixed point equation MA(f),A(g)(μ)=μ, where A(f) and A(g) are the quasiarithmetic means respectively generated by f and g. In this article, the operator MM,NMM,N is studied under less restrictive conditions and a general fixed point theorem is derived from an explicit formula for the iterates MnM,NMM,Nn . The concept of class of generalized Cauchy means associated to a given family of mixing pairs of means is introduced and some distinguished families of pairs are presented. The question of equality in these classes of means remains a challenging open problem.Fil: Berrone, Lucio Renato. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

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 $(\lambda_n)$ and $\mathscr{M}_{i=1}^n(x_i,\lambda_i)$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$, the inequality $$\mathscr{A}_{i=1}^n \big(\mathscr{M}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big) \le \mathscr{M}_{i=1}^n \big(\mathscr{A}_{j=1}^i (x_j,\lambda_j),\:\lambda_i\big)$$ holds for every $(x_n)$ and $(\lambda_n)$ such that the sequence $(\frac{\lambda_k}{\lambda_1+\cdots+\lambda_k})$ is decreasing.Comment: J. Inequal. Appl. (2018

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

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 $$\bigg(\frac{f}{g}\bigg)^{\!\!-1}\!\!\bigg(\frac{tf(x)+sf(y)}{tg(x)+sg(y)}\bigg) +\bigg(\frac{h}{k}\bigg)^{\!\!-1}\!\!\bigg(\frac{sh(x)+th(y)}{sk(x)+tk(y)}\bigg)=x+y \qquad(x,y\in I),$$ 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

Some functional equations related to the characterizations of information measures and their stability

The main purpose of this paper is to investigate the stability problem of some functional equations that appear in the characterization problem of information measures.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1307.0657, arXiv:1307.0631, arXiv:1307.0664, arXiv:1307.065

Prevention and contrast of child abuse and neglect in the practice of European paediatricians: a multi-national pilot study

Background: Child abuse and neglect, or maltreatment, is a serious public health problem, which may cause long-term effects on children's health and wellbeing and expose them to further adulthood vulnerabilities. Studies on child maltreatment performed in Europe are scarce, and the number of participants enrolled relatively small. The aim of this multi-national European pilot study, was to evaluate the level of understanding and perception of the concepts of child abuse and neglect by European paediatricians working in different medical settings, and the attitude toward these forms of maltreatment in their practice. Methods: The study was performed by a cross-sectional, descriptive, online survey, made available online to European paediatricians members of 50 national paediatric, who belonged to four different medical settings: hospital, family care, university centres and private practice. The questionnaire, designed as a multiple choice questions survey, with a single answer option consisted of 22 questions/statements. Frequency analyses were applied. Most of the data were described using univariate analysis and Chi-squared tests were used to compare the respondents and answers and a significance level of p ≤ 0.05 applied. Results: Findings show that European paediatricians consider the training on child maltreatment currently provided by medical school curricula and paediatric residency courses to be largely insufficient and continuing education courses were considered of great importance to cover educational gaps. Physical violence was recognized by paediatricians mostly during occasional visits with a significant correlation between detecting abuse during an occasional visit and being a primary care paediatrician. Results also showed a reluctance by paediatricians to report cases of maltreatment to the competent judicial authorities. Conclusions: Data of this study may provide useful contribution to the current limited knowledge about the familiarity of European paediatricians with child maltreatment and their skills to recognize, manage and contrast abusive childhood experiences in their practice. Finally, they could provide local legislators and health authorities with information useful to further improve public health approaches and rules able to effectively address shared risk and protective factors, which could prevent child abuse and neglect from ever occurring

Residual dipolar coupling investigation of a heparin tetrasaccharide confirms the limited effect of flexibility of the iduronic acid on the molecular shape of heparin

The solution conformation of a fully sulfated heparin-derived tetrasaccharide, I, was studied in the presence of a 4-fold excess of Ca2+. Proton–proton and proton–carbon residual dipolar couplings (RDCs) were measured in a neutral aligning medium. The order parameters of two rigid hexosamine rings of I were determined separately using singular value decomposition and ab initio structures of disaccharide fragments of I. The order parameters were very similar implying that a common order tensor can be used to analyze the structure of I. Using one order tensor, RDCs of both hexosamine rings were used as restraints in molecular dynamics simulations. RDCs of the inner iduronic acid were calculated for every point of the molecular dynamics trajectory. The fitting of the calculated RDCs of the two forms of the iduronic acid to the experimental values yielded a population of 1C4 and 2So conformers of iduronic acid that agreed well with the analysis based on proton–proton scalar coupling constants. The glycosidic linkage torsion angles in RDC-restrained molecular dynamics (MD) structures of I are consistent with the interglycosidic three-bond proton–carbon coupling constants. These structures also show that the shape of heparin is not affected dramatically by the conformational flexibility of the iduronic acid ring. This is in line with conclusions of previous studies based on MD simulations and the analysis of 1H-1H NOEs. Our work therefore demonstrates the effectiveness of RDCs in the conformational analysis of glycosaminoglycans

Estimation of interdomain flexibility of N-terminus of factor H using residual dipolar couplings

Characterization of segmental flexibility is needed to understand the biological mechanisms of the very large category of functionally diverse proteins, exemplified by the regulators of complement activation, that consist of numerous compact modules or domains linked by short, potentially flexible, sequences of amino acid residues. The use of NMR-derived residual dipolar couplings (RDCs), in magnetically aligned media, to evaluate interdomain motion is established but only for two-domain proteins. We focused on the three N-terminal domains (called CCPs or SCRs) of the important complement regulator, human factor H (i.e. FH1-3). These domains cooperate to facilitate cleavage of the key complement activation-specific protein fragment, C3b, forming iC3b that no longer participates in the complement cascade. We refined a three-dimensional solution structure of recombinant FH1-3 based on nuclear Overhauser effects and RDCs. We then employed a rudimentary series of RDC datasets, collected in media containing magnetically aligned bicelles (disk-like particles formed from phospholipids) under three different conditions, to estimate interdomain motions. This circumvents a requirement of previous approaches for technically difficult collection of five independent RDC datasets. More than 80% of conformers of this predominantly extended three-domain molecule exhibit flexions of < 40 °. Such segmental flexibility (together with the local dynamics of the hypervariable loop within domain 3), could facilitate recognition of C3b via initial anchoring and eventual reorganization of modules to the conformation captured in the previously solved crystal structure of a C3b:FH1-4 complex