Power means generated by some mean-value theorems

Demonstratio Mathematica, 2013

Some variants of the Lagrange and Cauchy mean-value theorems lead to the conclusion that means, in general, are not symmetric. They are symmetric iff they coincide (respectively) with the Lagrange and Cauchy means. Under some regularity assumptions, we determine the form of all the relevant symmetric means.

2004

If A is an isotonic linear functional and f : (a,b) ! (0,1) is a monotone function then Q(r,f) = (fr(a) + fr(b) A(fr))1/r is increasing in r.

Demonstratio Mathematica, 2010

Some generalizations of the Lagrange Mean-Value Theorem and Cauchy Mean-Value Theorem are proved and the extensions of the corresponding classes of means are presented.

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...

Bulletin of the Korean Mathematical Society, 2013

A mean-value result, saying that the difference quotient of a differentiable function in a real interval is a mean value of its derivatives at the endpoints of the interval, leads to the functional equation f (x) − F (y) x − y = M (g(x), G(y)) , x = y, where M is a given mean and f, F, g, G are the unknown functions. Solving this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homographic and square-root functions. A new criterion of the monotonicity of a real function is presented.

2003

Let M : (0, ∞) 2 → (0, ∞) be a homogeneous strict mean such that the function h := M (·, 1) is twice dierentiable and 0 = h (1) = 1. It is shown that if there exists an M -ane function, continuous at a point which is neither constant nor linear, then M must be a weighted power mean. Moreover the homogeneity condition of M can be replaced by M -convexity of two suitably chosen linear functions. With the aid of iteration groups, some generalizations characterizing the weighted quasi-arithmetic means are given. A geometrical aspect of these results is discussed.

Journal of Mathematical Analysis and Applications, 2006

We show that every Cauchy mean in (0, ∞) can be embedded into two parameter family of weighted means. Some basic properties and examples are presented. A functional equation which appears in the problem of symmetry of these means is considered. As an application a natural extension of Stolarsky's means is obtained and a two parameter subclass of weighted power means is determined.

International Journal of Mathematics and Mathematical Sciences, 2013

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class Λ f,g (a, b) of mean values where f, g are continuously differentiable convex functions satisfying the relation f ′′ (t) = tg ′′ (t), t ∈ R + . Then we asked for a characterization of f, g such that the inequalities H(a, b) ≤ Λ f,g (a, b) ≤ A(a, b) or L(a, b) ≤ Λ f,g (a, b) ≤ I(a, b) hold for each positive a, b, where H, A, L, I are the harmonic, arithmetic, logarithmic and identric means, respectively. For a subclass of Λ with g ′′ (t) = t s , s ∈ R, this problem is thoroughly solved.

Acta Mathematica Hungarica, 2010

Without any regularity conditions, we determine all the Cauchy means C [f,g] that are invariant with respect to the mean-type mapping (L [f ] , L [g]) where L [f ] denotes the Lagrangean mean generated by f. Applications in iteration theory and functional equation are given. n→∞

2020

Basing on the four types of Cauchy di erences, some general constructions of k-variable premeans and means generated by a single variable real function f de ned in a real interval I is discussed, special cases are examined and open questions are proposed. In particular, if I is closed under the addition, and f is such that F (x) := f (kx)−kf (x) is invertible, then the rst of four considered functions Mf : I k → R is of the form Mf (x1, ..., xk) = F −1 (f (x1 + ...+ xk)− (f (x1) + ...+ f (xk))) . Conditions under which Mf is a k-variable mean (referred to as quasiCauchy di erence mean of additive type) are examined.