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In this paper we study how the expectations of power means behave asymptotically as some relevant parameter approaches infinity and how to approximate them accurately for general nonnegative continuous probability distributions. We derive approximation formulae for such expectations as distribution mean increases, and apply them to some commonly used distributions in statistics and financial mathematics. By numerical computations we demonstrate the accuracy of the proposed formulae which behave well even for smaller sample sizes. Furthermore, analysis of behavior depending on sample size contributes to interesting connections with the power mean of probability distribution. [ABSTRACT FROM AUTHOR]

We aim to extend some recent refinements of the Jensen inequality. First, we establish even more precise estimates for the Jensen inequality. Then, we derive an improvement of the Jensen inequality with regard to h$$ h $$‐convexity. As an application, our main results are applied in deriving refinements of power mean inequalities and the Hölder inequality. Finally, we obtain more accurate estimates for f$$ f $$‐divergences, important in the information theory. [ABSTRACT FROM AUTHOR]

The focus of this paper is the application of the Fink identity in obtaining Jensen-type inequalities for higher order convex functions. In addition to the basic form, we establish superadditivity and monotonicity relations that correspond to the Jensen inequality in this setting. We also obtain the corresponding Lah-Ribarič inequality. The obtained results are valid for functions of even degree of convexity. With this method, we derive some new bounds for the differences of power means, as well as some new Hölder-type inequalities. [ABSTRACT FROM AUTHOR]

The main focus of this paper is a study of Jensen-type inequalities for the Lipschitzian functions. We establish the reverse of the Jensen inequality expressed in terms of the corresponding Lipschitz constant. In addition, we also obtain the reverse of the superadditivity relation for a convex function, expressed in the same way. As an application, we obtain reverses of power mean inequalities, the Hölder inequality, and the Hermite-Hadamard inequality, expressed in terms of the Lipschitzianity. In particular, we derive reverses of the arithmetic-geometric mean inequality in both difference and quotient forms. [ABSTRACT FROM AUTHOR]

We give a correction to the previous paper of S. F. Cao and T. T. Wang (2021). Additionally, provide a conversion formula of the power mean of a certain exponential sums weighted by a Dirichlet character χ³ and some of its applications. [ABSTRACT FROM AUTHOR]

In this paper we propose some Heinz-type inequalities for convex functions which provide a very simple proof of the main results of Kittaneh, Moslehian and Sababheh showed in the paper (F. Kittaneh, M. S. Moslehian and M. Sababheh, Quadratic interpolation of the Heinz means, Math. Inequal. Appl. 21(3) (2018) 739–757). We apply these inequalities to infer new inequalities for power means. As an application, we also give operator versions of Heinz-type inequalities for power and Heinz means. [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]

In this paper, we establish a new auxiliary identity of the Bullen type for twice-differentiable functions in terms of fractional integral operators. Based on this new identity, some generalized Bullen-type inequalities are obtained by employing convexity properties. Concrete examples are given to illustrate the results, and the correctness is confirmed by graphical analysis. An analysis is provided on the estimations of bounds. According to calculations, improved Hölder and power mean inequalities give better upper-bound results than classical inequalities. Lastly, some applications to quadrature rules, modified Bessel functions and digamma functions are provided as well. [ABSTRACT FROM AUTHOR]

In the present work, we prove a result concerning an ordering over intuitionistic fuzzy pairs generated by the power mean ( M p ) for p > 0 . We also introduce a family of orderings over intuitionistic fuzzy pairs generated by the weighted power mean ( M p α ) and prove that a similar result holds for them. The considered orderings in a natural way extend the classical partial ordering and allow the comparison of previously incomparable alternatives. In the process of proving these properties, we establish some inequalities involving logarithms which may be of interest by themselves. We also show that there exists p > 0 for which a finite set of alternatives, satisfying some reasonable requirements, some of which were not comparable under the classical ordering, has all its elements comparable under the new ordering. Finally, we provide some examples for the possible use of these orderings to a set of alternatives, which are in the form of intuitionistic fuzzy pairs as well as to results from InterCriteria Analysis. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the lower and upper bounds for the generalized Tsallis relative operator entropy which will recover some known results. Moreover, we verify the information monotonicity for the generalized Tsallis relative operator entropy and its complementary. In particular, this gives the information monotonicity for the Tsallis relative operator entropy and relative operator entropy. [ABSTRACT FROM AUTHOR]

Abstract Loewner partial order plays a very important role in metric topology and operator inequality on the open convex cone of positive invertible operators. In this paper, we consider a family G = { G n } n ∈ N of the ordered means for positive invertible operators equipped with homogeneity and properties related to the Loewner partial order such as the monotonicity, joint concavity, and arithmetic-G-harmonic weighted mean inequalities. Similar to the resolvent average, we construct a parameterized ordered mean and compare two types of mixtures of parameterized ordered means in terms of the Loewner order. We also show a relation between two families of parameterized ordered means associated with the power mean monotonic interpolating given two parameterized ordered means.

In this paper we present estimation formulas for the expectations of power means of large data and associate them with means of probability distribution and means of random sample. The proposed method follows from the asymptotic expansion of power means which is applicable for sufficiently large data and it is especially useful when value of such expectation is hard to obtain. We will show the accuracy of these approximations for random samples which have uniform and normal distribution and analyse their behaviour for large sample volume. [ABSTRACT FROM AUTHOR]

Convolutional Neural Network is one of the famous members of the deep learning family of neural network architectures, which is used for many purposes, including image classification. In spite of the wide adoption, such networks are known to be highly tuned to the training data (samples representing a particular problem), and they are poorly reusable to address new problems. One way to change this would be, in addition to trainable weights, to apply trainable parameters of the mathematical functions, which simulate various neural computations within such networks. In this way, we may distinguish between the narrowly focused task-specific parameters (weights) and more generic capability-specific parameters. In this paper, we suggest a couple of flexible mathematical functions (Generalized Lehmer Mean and Generalized Power Mean) with trainable parameters to replace some fixed operations (such as ordinary arithmetic mean or simple weighted aggregation), which are traditionally used within various components of a convolutional neural network architecture. We named the overall architecture with such an update as a hyper-flexible convolutional neural network. We provide mathematical justification of various components of such architecture and experimentally show that it performs better than the traditional one, including better robustness regarding the adversarial perturbations of testing data. [ABSTRACT FROM AUTHOR]

As generalization of Pythagorean means, several kinds of the power mean of positive definite Hermitian matrices have been studied. We introduce new power mean of positive definite operators as a fixed point of the equation modified from the operator Rényi divergence. We study its fundamental properties and compare it with other power means. [ABSTRACT FROM AUTHOR]

Motivated by some recent results known from the literature, in this paper, we establish a class of Jensen-type inequalities referring to functions of an even degree of convexity. The main idea of proving our results is a transformation of the classical Jensen functional via the Montgomery identity which is suitable to study in companion with the higher-order convexity. In such a way, we obtain superadditivity and monotonicity relations that correspond to the Jensen functional equipped with a function of an even degree of convexity. As an application, we obtain some new bounds for the differences of power means. Furthermore, we also establish some new Hölder-type inequalities. Finally, we study the Lah–Ribarič inequality in the above-described setting. [ABSTRACT FROM AUTHOR]

Loewner partial order plays a very important role in metric topology and operator inequality on the open convex cone of positive invertible operators. In this paper, we consider a family G = { G n } n ∈ N of the ordered means for positive invertible operators equipped with homogeneity and properties related to the Loewner partial order such as the monotonicity, joint concavity, and arithmetic-G-harmonic weighted mean inequalities. Similar to the resolvent average, we construct a parameterized ordered mean and compare two types of mixtures of parameterized ordered means in terms of the Loewner order. We also show a relation between two families of parameterized ordered means associated with the power mean monotonic interpolating given two parameterized ordered means. [ABSTRACT FROM AUTHOR]

In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen's inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater's inequality. The planned improvements are demonstrated in both discrete as well as in integral versions. The quoted results allow us to provide relationships for the power means. Moreover, with the help of established results, we present some estimates for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and Bhattacharyya coefficient. In addition, we discuss some additional applications of the main results for the Zipf–Mandelbrot entropy. [ABSTRACT FROM AUTHOR]

Suppose $w,v>0$, $w\neq v$ and $A_{u}\left (w,v\right) $ is the $u$-order power mean (PM) of $w$ and $v$. In this paper, we completely describe the convexity of $u\mapsto A_{u}\left (w,v\right) $ on $\mathbb{R}$ and $% s\mapsto A_{u\left (s\right) }\left (w,v\right) $ with $u\left (s\right) =\left (\ln 2\right) /\ln \left (1/s\right) $ on $\left (0,\infty \right) $. These yield some new inequalities for PMs, and give an answer to an open problem.

We provide an application of the theory of group actions to the study of musical scales. For any group G, finite G-set S, and real number t, we define the t-power diameter d i a m t (G , S) to be the size of any maximal orbit of S divided by the t-power mean orbit size of the elements of S. The symmetric group S 11 acts on the set of all tonic scales, where a tonic scale is a subset of Z 12 containing 0. We show that for all t ∈ [ − 1 , 1 ] , among all the subgroups G of S 11 , the t-power diameter of the G-set of all heptatonic scales is the largest for the subgroup Γ, and its conjugate subgroups, generated by { (1 2) , (3 4) , (5 6) , (8 9) , (10 11) }. The unique maximal Γ-orbit consists of the 32 thāts of Hindustani classical music popularized by Bhatkhande. This analysis provides a reason why these 32 scales, among all 462 heptatonic scales, are of mathematical interest. We also apply our analysis, to a lesser degree, to hexatonic and pentatonic scales. [ABSTRACT FROM AUTHOR]

This paper deals with the so-called generalized power-type Toader mean which is defined by ... for a,b > 0 with non-zero integer n. In this study, we establish the following chain of inequalities H(a,b) < T-1(a,b) < G(a,b) < T1(a,b) < A(a,b) < T2(a,b) < Q(a,b) < T3(a,b) < T4(a,b) < C(a,b) for all a,b > 0 with a = b, where H(a,b) = 2ab/(a + b), G(a,b) = √ab, A(a,b)=(a + b)/2, Q(a,b) = [(a² + b²)/2]1/2 and C(a,b)=(a² + b²)/(a+b) are the harmonic, geometric, arithmetic, quadratic and contra-harmonic means, respectively. Further, we provide sharp bounds for T-1(a,b) and T 4(a,b) in terms of bivariate means mentioned above. As applications, new bounds for complete elliptic integral of the second kind are established. [ABSTRACT FROM AUTHOR]

We consider in this paper probability measures with compact support on the open convex cone of positive definite Hermitian matrices. We define the least squares barycenter for the Bures–Wasserstein distance, called the Wasserstein barycenter, as a unique minimizer of the integral of squared Bures–Wasserstein distances. Furthermore, interesting properties of the Wasserstein barycenter including the iteration approach via optimal transport maps, the boundedness and extended Lie–Trotter–Kato formula, and several inequalities with power means have been established in the setting of compactly supported measures. [ABSTRACT FROM AUTHOR]

In the article, we prove that the double inequalities Ma1 (a,b) < Mtan(a,b) < Mß1 (a,b), Ma2 (a,b) < Msinh(a,b) < Mß2 (a,b) hold for all a,b > 0 with a = b if and only if a1 - 1/3, ß1 log2/log(2tan1) ? 0.61007, a2 x 2/3 and ß2 log2/log(2sinh1) ? 0.81109, where Mp, Mtan and Msinh are the pth power mean, tangent mean and hyperbolic sine mean, respectively. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove one inequality with power functions. A simplified form of the inequality was published as the problem 12024-02 in the American Mathematical Monthly.

In this paper, we use the analysis method and the properties of trigonometric sums to study the computational problem of one kind power mean of the hybrid Gauss sums. After establishing some relevant lemmas, we give an exact computational formula for it. As an application of our result, we give an exact formula for the number of solutions of one kind diagonal congruence equation mod p, where p be an odd prime.

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.

We study several properties of a family of quantum divergences with p -power means. We also show that the Hellinger distance with log Euclidean mean and the quantum Jensen-Shannon divergence can be rediscovered from our class of quantum distances via the limit processes. [ABSTRACT FROM AUTHOR]

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.

Recently, a new Riemannian metric and a least squares mean of positive definite matrices have been introduced. They are called the Bures-Wasserstein metric and Wasserstein mean, which are different from the Riemannian trace metric and Karcher mean. In this paper we find relationships of the Wasserstein mean with other matrix means such as the power means, harmonic mean, and Karcher mean. [ABSTRACT FROM AUTHOR]

The aggregation formula in the Human Development Index (HDI) was changed to a geometric mean in 2010. In this paper, we search for a theoretical justification for employing this new HDI formula. First, we find a maximal class of index functions, what we call quasi‐geometric means, that satisfy symmetry for the characteristics, normalization, and separability. Second, we show that power means are the only quasi‐geometric means satisfying homogeneity. Finally, the new HDI is the only power mean satisfying minimal lower boundedness, which is a local complementability axiom proposed by Herrero et al. (2010). [ABSTRACT FROM AUTHOR]

We present several Ando–Hiai type inequalities for n-variable operator means for positive invertible operators. Ando–Hiai's inequalities given here are not only of the original type but also of the complementary type and of the reverse type involving the generalized Kantorovich constant. [ABSTRACT FROM AUTHOR]

Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.

Abstract In this paper, we present the best possible parameters p and q such that the double inequality M p ( a , b ) < V ( a , b ) < M q ( a , b ) $$ M_{p}(a,b)< V(a,b)< M_{q}(a,b) $$ holds for all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , where M r ( a , b ) = [ ( a r + b r ) / 2 ] 1 / r $M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ ( r ≠ 0 $r\neq0$ ) and M 0 ( a , b ) = a b $M_{0}(a,b)= \sqrt {ab}$ is the rth power mean and V ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / 2 a b ) ] $V(a,b)=(a-b)/[\sqrt{2}\sinh^{-1}((a-b)/\sqrt{2ab})]$ is the second Yang mean.

A function ƒ : R+ → R+ is (m1, m2)-convex (concave) if ƒ(m1(x,y)) ≤ (≥) m2(ƒ(x), ƒ(y)) for all x,y Є R+ = (0,∞) and m1 and m2 are two mean functions. Anderson et al. [1] studies the dependence of (m1, m2)-convexity (concavity) on m1 and m2 and gave the sufficient conditions of (m1, m2)-convexity and concavity of a function defined by Maclaurin series. In this paper, we make a contribution to the topic and study the (m1, m2)-convexity and concavity of a function where m1 and m2 are identric mean, Alzer mean mean. As well, we prove a conjecture posed by Bruce Ebanks in [2].

After outlining the way in which an intrinsic mean G = { G n } on a complete metric space gives rise to a contractive barycentric map on some class of Borel probability measures and some basic examples of this process, we show how the resulting barycentric map gives rise to a general theory of integration of measurable functions into the space. We apply this machinery to the cone of positive invertible elements of a C ⁎ -algebra equipped with the Thompson metric to derive barycentric maps and their basic properties arising from the power means. Finally we derive basic results for the Karcher barycenter including its approximation by the barycentric maps for power means and its satisfaction of the Karcher equation. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the double inequality Mα(a,b) < NGQ(a,b) 0 with a ≠ b if and only if α ≤ 2log2/(5log2-2logπ) = 1:1785... and β ≥ 4/3,where NGQ(a,b) = [G(a,b)CQ2(a,b)=U(a,b)]/2 is the second Neuman mean, G(a,b) = √ab, Q(a,b) = √(a2+b2)/2 and U(a,b) = (a-b)/[√2tan-1((a - b)/ √2ab)] are the geometric, quadratic and Yang mean of a and b, respectively( [ABSTRACT FROM AUTHOR]

We derive algorithms for computing asymptotic expansion of the composite mean of two arbitrary means M and N . Then we analyse asymptotic behaviour of the compound mean M ⊗ N . Examples and application to some classical means are also presented. [ABSTRACT FROM AUTHOR]