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The symmetric Kullback–Leibler centroid, also called the Jeffreys centroid, of a set of mutually absolutely continuous probability distributions on a measure space provides a notion of centrality which has proven useful in many tasks, including information retrieval, information fusion, and clustering. However, the Jeffreys centroid is not available in closed form for sets of categorical or multivariate normal distributions, two widely used statistical models, and thus needs to be approximated numerically in practice. In this paper, we first propose the new Jeffreys–Fisher–Rao center defined as the Fisher–Rao midpoint of the sided Kullback–Leibler centroids as a plug-in replacement of the Jeffreys centroid. This Jeffreys–Fisher–Rao center admits a generic formula for uni-parameter exponential family distributions and a closed-form formula for categorical and multivariate normal distributions; it matches exactly the Jeffreys centroid for same-mean normal distributions and is experimentally observed in practice to be close to the Jeffreys centroid. Second, we define a new type of inductive center generalizing the principle of the Gauss arithmetic–geometric double sequence mean for pairs of densities of any given exponential family. This new Gauss–Bregman center is shown experimentally to approximate very well the Jeffreys centroid and is suggested to be used as a replacement for the Jeffreys centroid when the Jeffreys–Fisher–Rao center is not available in closed form. Furthermore, this inductive center always converges and matches the Jeffreys centroid for sets of same-mean normal distributions. We report on our experiments, which first demonstrate how well the closed-form formula of the Jeffreys–Fisher–Rao center for categorical distributions approximates the costly numerical Jeffreys centroid, which relies on the Lambert W function, and second show the fast convergence of the Gauss–Bregman double sequences, which can approximate closely the Jeffreys centroid when truncated to a first few iterations. Finally, we conclude this work by reinterpreting these fast proxy Jeffreys–Fisher–Rao and Gauss–Bregman centers of Jeffreys centroids under the lens of dually flat spaces in information geometry. [ABSTRACT FROM AUTHOR]

On the setting of positive definite operators we study the near-order properties of power means such as the quasi-arithmetic mean (Hölder mean) and Rényi power mean. We see the monotonicity of spectral geometric mean and Wasserstein mean on parameters with respect to the near-order and the near-order relationship between the spectral geometric mean and Wasserstein mean. Furthermore, the monotonicity of quasi-arithmetic mean on parameters and the convergence of Rényi power mean to the log-Euclidean mean with respect to the near-order have been established. [ABSTRACT FROM AUTHOR]

We present Hermite–Hadamard–Fejér type inequalities for strongly MφMψ -convex functions. Some refinements of them and bounds for the integral mean of the product of two functions are also obtained.

We present Hermite–Hadamard–Fejér type inequalities for strongly MφMψ -convex functions. Some refinements of them and bounds for the integral mean of the product of two functions are also obtained. [ABSTRACT FROM AUTHOR]

The symmetric Kullback–Leibler centroid, also called the Jeffreys centroid, of a set of mutually absolutely continuous probability distributions on a measure space provides a notion of centrality which has proven useful in many tasks, including information retrieval, information fusion, and clustering. However, the Jeffreys centroid is not available in closed form for sets of categorical or multivariate normal distributions, two widely used statistical models, and thus needs to be approximated numerically in practice. In this paper, we first propose the new Jeffreys–Fisher–Rao center defined as the Fisher–Rao midpoint of the sided Kullback–Leibler centroids as a plug-in replacement of the Jeffreys centroid. This Jeffreys–Fisher–Rao center admits a generic formula for uni-parameter exponential family distributions and a closed-form formula for categorical and multivariate normal distributions; it matches exactly the Jeffreys centroid for same-mean normal distributions and is experimentally observed in practice to be close to the Jeffreys centroid. Second, we define a new type of inductive center generalizing the principle of the Gauss arithmetic–geometric double sequence mean for pairs of densities of any given exponential family. This new Gauss–Bregman center is shown experimentally to approximate very well the Jeffreys centroid and is suggested to be used as a replacement for the Jeffreys centroid when the Jeffreys–Fisher–Rao center is not available in closed form. Furthermore, this inductive center always converges and matches the Jeffreys centroid for sets of same-mean normal distributions. We report on our experiments, which first demonstrate how well the closed-form formula of the Jeffreys–Fisher–Rao center for categorical distributions approximates the costly numerical Jeffreys centroid, which relies on the Lambert W function, and second show the fast convergence of the Gauss–Bregman double sequences, which can approximate closely the Jeffreys centroid when truncated to a first few iterations. Finally, we conclude this work by reinterpreting these fast proxy Jeffreys–Fisher–Rao and Gauss–Bregman centers of Jeffreys centroids under the lens of dually flat spaces in information geometry.

In this paper, we present the Sugeno integral of Hermite–Hadamard inequality for the case of quasi-arithmetically convex (q-ac) functions which acts as a generator for all quasi-arithmetic means in the frame work of Sugeno integral.

Let α 1 , … , α m be two or more positive reals with sum 1, let C ⊆ R k be an open convex set, and f : C → R k be a continuous injection with convex image. For each nonempty set S ⊆ C , let M (S) be the family of quasi-arithmetic means of all m-tuples of vectors in C with respect to f and the weights α 1 , … , α m , that is, the family M (S) = f - 1 α 1 f (x 1) + ⋯ + α m f (x m) : x 1 , … , x m ∈ S. We provide a simple necessary and sufficient condition on S for which the infinite iteration ⋃ n M n (S) is relatively dense in the convex hull of S. [ABSTRACT FROM AUTHOR]

We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is an example of a homogeneous quasi-arithmetic mean. Under certain conditions, several limit theorems hold for the power mean, similar to the case of the arithmetic mean of i.i.d. integrable random variables. Our feature is that the generators of the power means are allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. We establish integrabilities of the power mean of i.i.d. non-integrable random variables and a limit theorem for the variances of the power mean. We also consider the behavior of the power mean as the parameter of the power varies. The complex-valued power means are unbiased, strongly-consistent, robust estimators for the joint of the location and scale parameters of the Cauchy distribution. [ABSTRACT FROM AUTHOR]

Abstract A characterization of pairs of functions that can be separated by an M φ M ψ $M_{\varphi}M_{\psi}$ -convex function and related results are obtained. Also, a Hyers–Ulam stability result for M φ M ψ $M_{\varphi}M_{\psi}$ -convex functions is given.

A characterization of pairs of functions that can be separated by an M φ M ψ -convex function and related results are obtained. Also, a Hyers–Ulam stability result for M φ M ψ -convex functions is given. [ABSTRACT FROM AUTHOR]

Frequency histograms are ubiquitous, being practically used in any field of science. In this paper, we present a partial order for frequency histograms and, to our knowledge, no order of this kind has been yet defined. This order is based on the stochastic order of discrete probability distributions and it has invariance properties that make it unique. First, we model a frequency histogram as a sequence of bins associated with a discrete probability (or relative frequency) distribution. Then, we consider that two histograms are ordered if they are defined on the same sequence of bins and their respective frequency distributions are stochastically ordered. The ordering can be easily spotted because the respective cumulative distribution functions of the frequencies of two ordered histograms do not cross each other. Finally, with each bin we can associate a representative value of the bin, and for two ordered histograms it holds that all quasi-arithmetic means (such as arithmetic, harmonic, and geometric mean) of the representative values weighted by the frequencies are ordered in the same direction than the histograms are. Our theoretical study is supported by three experiments in the fields of image processing, traffic flow, and income distribution.

The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation f - 1 ( p 1 (x) f (x) + p 2 (y) f (y) p 1 (x) + p 2 (y) ) + g - 1 ( q 1 (x) g (x) + q 2 (y) g (y) q 1 (x) + q 2 (y) ) = x + y (x , y ∈ I) , where I is a nonempty open real interval and f , g : I → R are continuous, strictly monotone and p 1 , p 2 , q 1 , q 2 : I → R + are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of f, g, twice continuous differentiability of p1 and p2 and assuming that p1 differs from p2 on a dense subset of I, a necessary and sufficient condition for the equality above is that the unknown functions are of the form f = u v , g = w z , and p 1 q 1 = p 2 q 2 = v z , where u , v , w , z : I → R are arbitrary solutions of the second-order linear differential equation F ′ ′ = γ F (γ ∈ R is arbitrarily fixed) such that v > 0 and z > 0 holds on I and { u , v } and { w , z } are linearly independent. [ABSTRACT FROM AUTHOR]

We show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function F : I 2 → I is continuous. As a consequence, we obtain a finer characterization of quasi-arithmetic means than the classical results of Aczél [1], Kolmogoroff [18], Nagumo [20] and de Finetti [11]. [ABSTRACT FROM AUTHOR]

Abstract In this article, we prove that the double inequalities α1[7C(a,b)16+9H(a,b)16]+(1−α1)[3A(a,b)4+G(a,b)4]0 $a, b>0$ with a≠b $a\neq b$ if and only if α1≤3/16=0.1875 $\alpha_{1}\leq 3/16=0.1875$, β1≥64/π2−6=0.484555… $\beta_{1}\geq64/\pi^{2}-6= 0.484555\dots$, α2≤3/16=0.1875 $\alpha_{2}\leq3/16=0.1875$ and β2≥(5log2−log3−2logπ)/(log7−log6)=0.503817… $\beta_{2}\geq(5\log2-\log3-2\log \pi)/(\log7-\log6)= 0.503817\dots$, where E(a,b)=(2π∫0π/2acos2θ+bsin2θdθ)2 $E(a,b)= (\frac{2}{\pi}\int^{\pi/2}_{0}\sqrt{a\cos^{2}\theta +b\sin^{2}\theta}\,d\theta )^{2}$, H(a,b)=2ab/(a+b) $H(a,b)=2ab/(a+b)$, G(a,b)=ab $G(a,b)=\sqrt{ab}$, A(a,b)=(a+b)/2 $A(a,b)=(a+b)/2$ and C(a,b)=(a2+b2)/(a+b) $C(a,b)=(a^{2}+b^{2})/(a+b)$ are the quasi-arithmetic, harmonic, geometric, arithmetic and contra-harmonic means of a and b, respectively.

In this paper we give a new refinement of the Lah–Ribarič inequality and, using the same technique, we give a refinement of the Jensen inequality. Using these results, a refinement of the discrete Hölder inequality and a refinement of some inequalities for discrete weighted power means and discrete weighted quasi-arithmetic means are obtained. We also give applications in the information theory; namely, we give some interesting estimations for the discrete Csiszár divergence and for its important special cases.

Abstract In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean M $\mathscr{M}$ the Kedlaya-type inequality A(x1,M(x1,x2),…,M(x1,…,xn))≤M(x1,A(x1,x2),…,A(x1,…,xn)) $$ \mathscr{A} \bigl(x_{1},\mathscr{M}(x_{1},x_{2}), \ldots,\mathscr{M}(x _{1},\ldots,x_{n}) \bigr) \le \mathscr{M} \bigl( x_{1}, \mathscr{A}(x _{1},x_{2}), \ldots,\mathscr{A}(x_{1},\ldots,x_{n}) \bigr) $$ holds for an arbitrary (xn) $(x_{n})$ ( A $\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if (xn) $(x_{n})$ is a vector with corresponding (non-normalized) weights (λn) $(\lambda_{n})$ and Mi=1n(xi,λi) $\mathscr{M}_{i=1}^{n}(x _{i},\lambda_{i})$ denotes the weighted mean then, under analogous conditions on M $\mathscr{M}$, the inequality holds for every (xn) $(x_{n})$ and (λn) $(\lambda_{n})$ such that the sequence (λkλ1+⋯+λk) $(\frac{\lambda_{k}}{\lambda_{1}+\cdots+\lambda_{k}})$ is decreasing.

The purpose of this paper is to investigate the equality problem of generalized Bajraktarević means, i.e., to solve the functional equation * f (- 1) ( p 1 (x 1) f (x 1) + ⋯ + p n (x n) f (x n) p 1 (x 1) + ⋯ + p n (x n) ) = g (- 1) ( q 1 (x 1) g (x 1) + ⋯ + q n (x n) g (x n) q 1 (x 1) + ⋯ + q n (x n) ) , which holds for all (x 1 , ⋯ , x n) ∈ I n , where n ≥ 2 , I is a nonempty open real interval, the unknown functions f , g : I → R are strictly monotone, f (- 1) and g (- 1) denote their generalized left inverses, respectively, and p = (p 1 , ⋯ , p n) : I → R + n and q = (q 1 , ⋯ , q n) : I → R + n are also unknown functions. This equality problem in the symmetric two-variable (i.e., when n = 2 ) case was already investigated and solved under sixth-order regularity assumptions by Losonczi (Aequationes Math 58(3):223–241, 1999). In the nonsymmetric two-variable case, assuming the three times differentiability of f, g and the existence of i ∈ { 1 , 2 } such that either p i is twice continuously differentiable and p 3 - i is continuous on I, or p i is twice differentiable and p 3 - i is once differentiable on I, we prove that (*) holds if and only if there exist four constants a , b , c , d ∈ R with a d ≠ b c such that c f + d > 0 , g = a f + b c f + d , and q ℓ = (c f + d) p ℓ (ℓ ∈ { 1 , ⋯ , n }). In the case n ≥ 3 , we obtain the same conclusion with weaker regularity assumptions. Namely, we suppose that f and g are three times differentiable, p is continuous and there exist i , j , k ∈ { 1 , ⋯ , n } with i ≠ j ≠ k ≠ i such that p i , p j , p k are differentiable. [ABSTRACT FROM AUTHOR]

Given two continuous functions f , g : I → 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 × T → I , and a probability measure μ on the measurable sets A , the d-variable mean M f , g , m ; μ : I d → I is defined by M f , g , m ; μ (x) : = (f g ) - 1 ( ∫ T f (m (x 1 , ... , x d , t)) d μ (t) ∫ T g (m (x 1 , ... , x d , t)) d μ (t) ) (x = (x 1 , ... , x d) ∈ 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 (x) = M h , k , m ; μ (x) (x ∈ I d) and the homogeneity property M f , g , m ; μ (λ x) = λ M f , g , m ; μ (x) (λ > 0 , x , λ x ∈ I d) , respectively, be satisfied. [ABSTRACT FROM AUTHOR]

This paper offers a solution of the functional equation (t f (x) + (1 - t) f (y)) φ (t x + (1 - t) y) = t f (x) φ (x) + (1 - t) f (y) φ (y) (x , y ∈ I) , where t ∈ ] 0 , 1 [ , φ : I → R is strictly monotone, and f : I → R is an arbitrary unknown function. As an immediate application, we shed new light on the equality problem of Bajraktarević means with quasi-arithmetic means. [ABSTRACT FROM AUTHOR]

The aim of this paper is to find a broad family of means defined on a subinterval of $I\subset [0,+\infty)$ such that $$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }\mathscr{M}(a_{1},\ldots ,a_{n}) Equivalently, the averaging operator $(a_{1},\,a_{2},a_{3}\,,\ldots)\mapsto (a_{1},\,\mathscr{M}(a_{1},a_{2}),\,\mathscr{M}(a_{1},a_{2},a_{3}),\ldots)$ is a selfmapping of $\ell _{1}(I)$. This property is closely related to the so-called Hardy inequality for means (which additionally requires boundedness of this operator). We prove that these two properties are equivalent in a broad family of so-called Gini means. Moreover, we show that this is not the case for quasi-arithmetic means, that is functions $f^{-1}(\sum f(a_{i})/n)$ , where $f:I\rightarrow \mathbb{R}$ is continuous and strictly monotone, $n\in \mathbb{N}$ and $a\in I^{n}$. However, the weak Hardy property is localisable for this family. [ABSTRACT FROM AUTHOR]

Abstract In the article, we present the best possible parameters λ = λ ( p ) $\lambda=\lambda (p)$ and μ = μ ( p ) $\mu=\mu(p)$ on the interval [ 0 , 1 / 2 ] $[0, 1/2]$ such that the double inequality G p [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] A 1 − p ( a , b ) < E ( a , b ) < G p [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] A 1 − p ( a , b ) $$\begin{aligned}& G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a \bigr]A^{1-p}(a,b) \\& \quad< E(a,b) < G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}$$ holds for any p ∈ [ 1 , ∞ ) $p\in[1, \infty)$ and all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , where A ( a , b ) = ( a + b ) / 2 $A(a, b)=(a+b)/2$ , G ( a , b ) = a b $G(a,b)=\sqrt{ab}$ and E ( a , b ) = [ 2 ∫ 0 π / 2 a cos 2 θ + b sin 2 θ d θ / π ] 2 $E(a,b)=[2\int_{0}^{\pi /2}\sqrt{a\cos^{2}\theta+b\sin^{2}\theta}\,d\theta/\pi]^{2}$ are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

Abstract In this paper we present some characterizations for quasi-arithmetic operator means (among them the arithmetic and harmonic means) on the positive definite cone of the full algebra of Hilbert space operators, and also for the Kubo–Ando geometric mean on the positive definite cone of a general non-commutative C ⁎ -algebra. [ABSTRACT FROM AUTHOR]

Abstract T -generable indistinguishability operators are operators E that can be expressed in the form E = T (E μ 1 , E μ 2 ,... , E μ m ) , where T is a t -norm and E μ is the fuzzy relation generated by the fuzzy subset μ. In this paper we analyse their relation with powers with respect to the t -norm T and with quasi-arithmetic means. For non-strict continuous Archimedean t-norms they are completely characterised as generable by crisp equivalence relations. These fuzzy relations are used to define a method, called JADE , useful for feature selection and classification tasks. JADE is based on minimising the distance between two indistinguishability measures: the one given by weighting the attribute-values describing the domain objects and the other one given by the correct classification taken as an equivalence relation. The preliminary experiments we carried out with JADE are promising concerning the accuracy in solving classification tasks. We also report some issues of the method that could be improved in the future. [ABSTRACT FROM AUTHOR]

Two years ago, during the 21st European Conference on Iteration Theory, Gregory Derfel asked: "Does there exist a non-trivial bounded continuous solution of the equation 2f(x)=f(x-1)+f(-2x)?" He repeated the question during the 55th International Symposium on Functional Equations. In this paper we present a partial solution of a more general problem, connected to the functional equation f(x)=M(f(x+t1),f(x+t2),...,f(x+tn-1),f(ax)), where n∈N,t1,t2,...,tn-1∈R\{0},a∈(-∞,0) and M is a given function in n variables satisfying some additional properties. In particular, M can be a weighted quasi-arithmetic mean in n variables. [ABSTRACT FROM AUTHOR]

It is known that the quasi-arithmetic means can be characterized in various ways, with an essential role of a symmetry property. In the expected utility theory, the quasi-arithmetic mean is called the certainty equivalent and it is applied, e.g., in a utility-based insurance contracts pricing. In this paper, we introduce and study the quasi-arithmetic type mean in a more general setting, namely with the expected value being replaced by the generalized Choquet integral. We show that a functional that is defined in this way is a mean. Furthermore, we characterize the equality, positive homogeneity, and translativity in this class of means.

We generalize a theorem proved by R. Ger and T. Kochanek [4]. Hence we obtain the solution of a problem posed by Z. Daróczy, J. Jarczyk and W. Jarczyk [2]. [ABSTRACT FROM AUTHOR]

Since positive definite Hermitian matrices have become fundamental objects in many areas, a variety of theoretical and computational research topics have been arisen. Especially, the average of positive definite matrices is a very important notion to see the central tendency of objects. There are many different kinds of averages for a finite number of positive definite matrices such as quasi-arithmetic means, power means and Cartan barycenters. We generalize these averages to the setting of positive definite matrices equipped with probability measures of compact support, and show the monotonicity of quasi-arithmetic means for parameters ≥ 1 {\geq 1} , and connections with inequalities between quasi-arithmetic means and power means, and between quasi-arithmetic means and Cartan barycenters. [ABSTRACT FROM AUTHOR]

The Jensen–Shannon divergence is a renowned bounded symmetrization of the unbounded Kullback–Leibler divergence which measures the total Kullback–Leibler divergence to the average mixture distribution. However, the Jensen–Shannon divergence between Gaussian distributions is not available in closed form. To bypass this problem, we present a generalization of the Jensen–Shannon (JS) divergence using abstract means which yields closed-form expressions when the mean is chosen according to the parametric family of distributions. More generally, we define the JS-symmetrizations of any distance using parameter mixtures derived from abstract means. In particular, we first show that the geometric mean is well-suited for exponential families, and report two closed-form formula for (i) the geometric Jensen–Shannon divergence between probability densities of the same exponential family; and (ii) the geometric JS-symmetrization of the reverse Kullback–Leibler divergence between probability densities of the same exponential family. As a second illustrating example, we show that the harmonic mean is well-suited for the scale Cauchy distributions, and report a closed-form formula for the harmonic Jensen–Shannon divergence between scale Cauchy distributions. Applications to clustering with respect to these novel Jensen–Shannon divergences are touched upon.

In this paper, we are going to describe the solutions of the functional equation concerning the unknown functionsandfdefined on an open interval. In our main result only the continuity of the functionand a regularity property of the set of zeros offare assumed. As application, we determine the solutions of the functional equation under monotonicity and differentiability conditions on the unknown functionsF, G, H, g, h. [ABSTRACT FROM PUBLISHER]

Given two continuous functions f , g : I → 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 × T → I , and a probability measure μ on the measurable sets A , the d -variable mean M f , g , m ; μ : I d → I is defined by M f , g , m ; μ ( x ) : = ( f g ) − 1 ( ∫ T f ( m ( x 1 , … , x d , t ) ) d μ ( t ) ∫ T g ( m ( x 1 , … , x d , t ) ) d μ ( t ) ) ( x = ( x 1 , … , x d ) ∈ 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 ( x ) ≤ M h , k , n ; ν ( x ) ( x ∈ I d ) be satisfied. [ABSTRACT FROM AUTHOR]

In the article, we present the best possible parameters $\lambda=\lambda (p)$ and $\mu=\mu(p)$ on the interval $[0, 1/2]$ such that the double inequality holds for any $p\in[1, \infty)$ and all $a, b>0$ with $a\neq b$ , where $A(a, b)=(a+b)/2$ , $G(a,b)=\sqrt{ab}$ and $E(a,b)=[2\int_{0}^{\pi /2}\sqrt{a\cos^{2}\theta+b\sin^{2}\theta}\,d\theta/\pi]^{2}$ are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively. [ABSTRACT FROM AUTHOR]

A T -indistinguishability operator (or fuzzy similarity relation) E is called unidimensional when it may be obtained from one single fuzzy subset (or fuzzy criterion). In this paper, we study when a T -indistinguishability operator that has been obtained as an average of many unidimensional ones is unidimensional too. In this case, the single fuzzy subset used to generate E is explicitly obtained as the quasi-arithmetic mean of all the fuzzy criteria primarily involved in the construction of E . [ABSTRACT FROM AUTHOR]

Popoviciu's inequality is extended to the framework of h-convexity and also to convexity with respect to a pair of quasi-arithmetic means. Several applications are included. [ABSTRACT FROM AUTHOR]