On geometric construction of some power means

European Journal of Applied Mathematics, 2008

We study a scale of two-component composite structures of equal proportions with infinitely many microlevels. The structures are obtained recursively and we find that their effective conductivities are power means of the local conductivities.

2021

Means of positive numbers appear in many applications and have been a traditional matter of study. In this work, we focus on defining a new mean of two positive values with some properties which are essential in applications, ranging from subdivision and multiresolution schemes to the numerical solution of conservation laws. In particular, three main properties are crucial—in essence, the ideas of these properties are roughly the following: to stay close to the minimum of the two values when the two arguments are far away from each other, to be quite similar to the arithmetic mean of the two values when they are similar and to satisfy a Lipchitz condition. We present new means with these properties and improve upon the results obtained with other means, in the sense that they give sharper theoretical constants that are closer to the results obtained in practical examples. This has an immediate correspondence in several applications, as can be observed in the section devoted to a par...

Sitzungsberichte und Anzeiger, 2005

In this note we introduce a one-parameter family of homogeneous means strictly related to ellipses. Each member of the family is a weighted power mean, and only one of them is both symmetric and quasi-arithmetic. Geometric interpretations are given, and higher-dimensional counterparts of these means are defined. Iterations of some mean-type mappings and some functional equations are considered.

2003

This operator appears, for example, in the time invariant linear optimal regulator system: Indeed, it is well known that, (Ref [1]), the particular algebraic Riccati equation, XAX = B has one and only one symmetric positive solution given by X = A−1(g)B. Very recently, Atteia and Raissouli (Ref [1]), have constructed the geometric mean of two convex functionals from a sequence of iterations descended from the sum and the infimal convolution. This approach permitted the authors of [1] to deduce another definition of the geometric operator mean which, of course, coincides with the given above one. Moreover, a physical illustration of the geometric mean as an equivalent resistor of an electrical circuit with matrices elements is introduced in [1].

Lobachevskii Journal of Mathematics, 2019

Bhatia, Lim, and Yamazaki conjectured that the Kubo-Ando extensions of means of numbers satisfy a norm minimality condition with respect to unitarily invariant norms. In this short note, we introduce a symmetric Kubo-Ando mean and a non-Kubo-Ando extension that do not satisfy this property.

Proceedings of the American Mathematical Society, 2011

According to a new mean-value theorem, under the conditions of a function f ensuring the existence and uniqueness of Lagrange's mean, there exists a unique mean M such that f (x) − f (y) x − y = M f (x), f (y). The main result says that, in this equality, M is a power mean if, and only if, M is either geometric, arithmetic or harmonic. A Cauchy relevant type result is also presented.

Nonlinear Analysis: Theory, Methods & Applications, 1994

For a given amount m of mass, we study the class of materials which can be reached by homogenization distributing the mass m on periodic structures of prescribed dimension k n in R n . Both in the scalar case of conductivity and in the vectorvalued case of elasticity, we find some bounds for the effective coefficients, depending on the mass m and the dimension parameters k, n. In the scalar case we prove that such bounds are optimal, as they do describe the set of all materials reachable by homogenization of structures of the type under consideration; in the vector-valued case we show that some of our estimates are attained.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Linear Algebra and its Applications, 2011

We define a new family of matrix means {L μ (ω; A)} μ∈R where ω and A vary over all positive probability vectors in R m and m-tuples of positive definite matrices resp. Each of these means interpolates between the weighted harmonic mean (μ = −∞) and the arithmetic mean of the same weight (μ = ∞) with L μ L ν for μ ν.

Bulletin of the Malaysian Mathematical Sciences Society, 2015

The aim of a study of the presented paper is the differential subordination involving harmonic means of the expressions p(z), p(z) + zp (z), and p(z) + zp (z) p(z) when p is an analytic function in the unit disk, such that p(0) = 1, p(z) ≡ 1. Several applications in the geometric functions theory are given.