Janusz Matkowski

Means and their inequalities

The theory of means has its roots in the work of the Pythagoreans who introduced the harmonic, geometric, and arithmetic means with reference to their theories of music and arithmetic. Later, Pappus introduced seven other means and gave the well-known elegant geometric proof of the celebrated inequalities among the harmonic, geometric, and arithmetic means. Nowadays, the families and types of means that are being investigated by researchers and the variety of questions that are being asked about them are beyond the scope of any single survey, with the voluminous book Handbook of Means and Their Inequalities by P. S. Bullen being the best such reference in this direction. The theory of means has grown to occupy a prominent place in mathematics with hundreds of papers on the subject appearing every year. The strong relations and interactions of the theory of means with the theories of inequalities, functional equations, and probability and statistics add greatly to its importance.

The converse of the Minkowski’s inequality theorem and its generalization

Let (£2, X, p.) be a measure space with two sets A, B el. such that 0 :R+ -+ R+ be bijective and (i> continuous at 0. We prove that if for all //-integrable step functions JCy:fi->R, ~X (I 1 . In the case of normalized measure we prove a generalization of Minkowskis inequality theorem. The suitable results for the reversed inequality are also presented. Introduction Let (Cl, 2, p.) be a measure space. The celebrated Minkowskis inequality theorem may be formulated in the form of the following implication. If cp(t) = cp(\)f (t>0), where p > 1 and cp(\) > 0, then (1) cp~x (I cpo\x + y\dp\ < cp~x (j <po\x\dp) + cp~x (I cp o \y\dp\ for every x and y belonging to the linear space of all the p-integrable step functions. This is a weaker version of the classical result which asserts that inequality (1) holds for all x and y from the Banach space Lp(Cl, X, p). But on the one hand the classical result is a simple consequence of the above implication, on the other hand the weaker version is more suitable for our purposes. In this paper we give conditions under which the converse implication holds. It follows from Mulhollands inequality [5] (cf. also M. Kuczma [3, p. 201, Theorem 1]) that, in general, the converse implication is false. (A simple example Received by the editors September 26, 1988 and, in revised forms, February 4, 1989 and August 1, 1989; a partial result has been presented to the 26th ISFE, Sant Felieu de Guixols, Spain, April 24-May 3, 1988. The contents of this paper have been presented to the Northeast Conference on General Topology and Applications, New York, June 15-17, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E30, 26D15, 26D10, 39C05; Secondary 39B40.

Nonlinear Contractions on Semimetric Spaces

Abstract Let (X, d) be a Hausdorff semimetric (d need not satisfy the triangle inequality) and d–Cauchy complete space. Let ƒ be a selfmap on X, for which d(ƒx, ƒy) ≤ φ(d(x, y)), (x, y ∈ X), where φ is a non– decreasing function from R +, the nonnegative reals, into R + such that φn (t) → 0, for all t ∈ R +. We prove that ƒ has a unique fixed point if there exists an r > 0, for which the diameters of all balls in X with radius r are equi-bounded. Such a class of semimetric spaces includes the Frechet spaces with a regular ecart, for which the Contraction Principle was established earlier by M. Cicchese [Boll. Un. Mat. Ital 13–A: 175-179, 1976], however, with some further restrictions on a space and a map involved. We also demonstrate that for maps ƒ satisfying the condition d(ƒx, ƒy) ≤ φ(max{d(x, ƒx), d(y, ƒy)}), (x, y ∈ X) (the Bianchini [Boll. Un. Mat. Ital. 5: 103–108, 1972] type condition), a fixed point theorem holds under substantially weaker assumptions on a distance function d.

On subadditive functions

The main result says that every one-to-one subadditive function f: (0, oo) -* (0, oo) such that limt,0 f(t) = 0 must be continuous everywhere. A construction of a broad class of discontinuous subadditive bijections of (0, oo) which are bounded in every vicinity of 0 is given. Moreover, a problem of extension of a subadditive function defined in (0, oo) to a subadditive even function in JR is considered

On Mulholland’s inequality

H.P. Mulholland has presented a sufficient condition for a generalization of the Minkowski inequality and another such condition was given by R.M. Tardiff. We show that Mulhollands condition implies Tardiffs, but that the converse is false.

An invariance of the geometric mean with respect to Stolarsky mean-type mappings

We determine all pairs of Stolarsky means (Er,s, Ek,m) such that G o (Er,s, Ek,m) = G, where G = E0,0 is the geometric mean. The convergence of the sequences of iterates of the mean-type mapping (Er,s, Ek,m) to the mapping (G, G) is considered. An application to a functional equation is given.

On Invariant Generalized Beckenbach-Gini Means

A functional equation that characterizes generalized Beckenbach-Gini means which are invariant with respect to Beckenbach-Gini mean-type mappings is considered. In the case when an invariant mean is either arithmetic or geometric or harmonic, without any regularity conditions, all solutions are found. In the general case, under some regularity assumptions, a necceasary condition is given. For positively homogeneous Beckenbach-Gini means a complete list of solutions is established. Translative Beckenbach-Gini means are also examined.

Quasi-monotonicity, subadditive bijections of R+, and characterization of Lp-norm

Abstract Real functions with a regular behaviour of the one-sided upper and lower limits are considered. The main result says that every subadditive bijection of R + right continuous at 0, must be a homeomorphism of R +. An application to a characterization of Lp-norm is also given.

The converse theorem for Minkowski's inequality

Abstract Let (Ω, Σ, μ) be a measure space and ϕ, ψ : (0, ∞) → (0, ∞) some bijective functions. Suppose that the functional P ϕ,ψ defined on class of μ-integrable simple functions χ : Ω → [0, ∞), μ({ϖ : χ(ϕ) > 0} > 0, by the formula P ϕ,ψ (χ) = ψ ∫ {χ>0} ϕo χdμ satisfies the triangle inequality. We prove that if there are A, B ϵ Σ such that 0 ϕ(t)=ϕ(1)t p , ψ(t)=ψ(1)t 1 p , t>0) , The assumption limt → 0 ψ(t) = 0 can be significantly weakened or, for some measure spaces, even omitted. The remaining assumptions are essential. In particular, in each of the cases: (i) A ϵ Σ ⇒ μ(A) = 0 or μ(A ≥ 1; (ii) A ϵ Σ ⇒ μ(A) ≤ 1 or μ(A) = ∞, some broad classes of pairs (ϕ, ψ) of non-power functions for which P ϕ,ψ is subadditive are indicated. These results give a solution of an open problem posed by W. Wnuk. The reversed triangle inequality is also considered.

On The Polynomial-Like Iterative Functional Equation

in this paper we give some properties of solutions of the iterative functional f n+1 (x)+⋯+a 0 x, equation considering its characteristic equation. Auseful method to discuss the general case is detailed described for the case n = 2.