GENERALIZED ABSTRACTED MEAN VALUES

2001

We develop a parallel theory to that concerning the concept of integral mean value of a function, by replacing the additive framework with a multiplicative one. Particularly, we prove results which are multiplicative analogues of the Jensen and Hermite-Hadamard inequalities.

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.

Journal of Optimization Theory and Applications, 1977

Properties of generalized convex functions, defined in terms of the generalized means introduced by Hardy, Littlewood, and Polya, are easily obtained by showing that generalized means and generalized convex functions are in fact ordinary arithmetic means and ordinary convex functions, respectively, defined on linear spaces with suitably chosen operations of addition and multiplication. The results are applied to some problems in statistical decision theory.

Mathematics, 2020

We prove that whenever the selfmapping (M1,…,Mp):Ip→Ip, (p∈N and Mi-s are p-variable means on the interval I) is invariant with respect to some continuous and strictly monotone mean K:Ip→I then for every nonempty subset S⊆{1,…,p} there exists a uniquely determined mean KS:Ip→I such that the mean-type mapping (N1,…,Np):Ip→Ip is K-invariant, where Ni:=KS for i∈S and Ni:=Mi otherwise. Moreover min(Mi:i∈S)≤KS≤max(Mi:i∈S). Later we use this result to: (1) construct a broad family of K-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.

Results in Mathematics, 1990

We study a certain monotonicity property of ratios of means, which we call a strong inequality. These strong inequalities were recently shown to be related to the so-called relative metric. We also use the strong inequalities to derive new ordinary inequalities. The means studied are the extended mean value of Stolarsky, Gini's mean and Seiffert's mean. , 26D05. increasing mean (of two variables) we understand a symmetric function M : R > × R > → R > which satisfies min{x, y} ≤ M (x, y) ≤ max{x, y} and M (sx, sy) = sM (x, y) for all s, x, y ∈ R > and for which t M (x) := M (x, 1) is increasing for x ∈ [1, ∞). The function t M is called the trace of M and uniquely determines M , since M (x, y) = yt M (x/y). If M and N are symmetric homogeneous increasing means we say that M is strongly greater than or equal to N , M N , if t M t N .

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.

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.

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.

ISRN Mathematical Analysis, 2011

We extend the classical notions of translativity and homogeneity of means to F-homogeneity, that is, invariance with respect to an operation F : I × I → I. We find the shape of F for the arithmetic weighted mean and then the general form of F for quasi-linear means. Also, we are interested in characterizations of means. It turns out that no quasi-arithmetic mean can be characterized by F-homogeneity with respect to a single operation F, one needs to take two of such operations in order to characterize a mean.