Beta-type functions and the harmonic mean

For arbitrary f:a,∞→0,∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\left( a,\infty \right) \rightarrow \left( 0,\infty \right) ,$$\end{document}a≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\ge 0,$$\end{document} the bivariable function Bf:a,∞2→0,∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{f}:\left( a,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) ,$$\end{document} related to the Euler Beta function, is considered. It is proved that Bf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{f\text { }}$$\end{document}is a mean iff it is the harmonic mean H. Some applications to the theory of iterative functional equations are given.

In this paper we are interested in answering when a beta-type function is a bivariable mean in (a, ∞). Our main result says that the beta-type function of a generator f : (a, ∞) → (0, ∞) is a mean iff f (x) = 2xe α(x) where α : R → R is an additive function, or equivalently, that B f is the harmonic mean (Theorem 2). This substantially improves the result of [1] where the homogeneity of the beta-type function is assumed.
In the preliminary Sect. 2 we recall the notions of mean, premean, reflexivity of a function, and some of their properties. The increasingness of the beta-type function B f is equivalent to the concavity of the function log •f in the sense of Wright (Proposition 1). In Sect. 3  . At the end we propose a unique and natural extension of the harmonic bivariable mean to R 2 .
The case of k-variable beta-type functions, k ≥ 3, will be considered in our next paper.
the beta-type function of generator f .
Note that the beta-type function B f of a generator f : In this context one could also consider the functions of beta-type of the generators f defined on the intervals (−∞, a) with values in (−∞, 0) .

Remark 4.
Replacing f in this Definition 3 by 1 f we get We shall prove the following be a continuous function. The following two conditions are equivalent: Proof. Assume (i). Then, as log is increasing in (0, ∞) , the function log •B f is increasing in each variable. So, by the definition of B f , for all x, y, z ∈ (0, ∞), or, equivalently, for all x, y, z ∈ (0, ∞), Choosing arbitrary u, v > 0, u < v, and t ∈ (0, 1), and taking we obtain that the above implication is equivalent to the following one: for all u, v > 0, and t ∈ (0, 1), which shows that B f is increasing if, and only if, log •f is concave in the sense of Wright. Since f is continuous, in view of a theorem of Ng [6], log•f is Wright concave if, and only if, log •f is concave. Since B f is a mean, it is reflexive. Thus we have shown that (ii) holds true. The converse implication follows from Remark 3.

Beta-type functions and reflexivity
Applying the method of the theory of iterative functional equations by Kuczma [4], we prove the following 2a) . The function f satisfies the functional equation if, and only if, 2a) . The function f satisfies the functional equation
(ii) By induction, we shall prove that, for all n ∈ N 0 , Take arbitrarily x ∈ (0, 1) . There exists a unique n ∈ N 0 such that If n = 0, then 1 ≤ x < 2 and, by the definition of f 0 , so (3.10) holds true for n = 0. Assume that (3.10) holds true for some n ∈ N 0 .
Taking arbitrarily x ∈ 1 2 n+1 , 1 2 n , we have 2x ∈ 1 2 n , 1 2 n−1 , thus, applying (3.10), we obtain Hence, by (3.3), so (3.10) holds true for x ∈ 1 2 n+1 , 1 2 n , which means that (3.10) holds for n + 1. By the induction principle, formula (3.10) holds true for all x ∈ (0, 1). Both reasonings prove the validity of (3.4), which is the second statement of our theorem. Arguing similarly as in the previous case we can prove the converse of (ii). In part (iii), since one implication is obvious, we must show that the continuity of f 0 and (3.5) imply the continuity of f . By (3.4), the continuity of f 0 on (1, 2) implies that f is continuous on n∈Z 2 n , 2 n+1 . It remains to show the continuity of f at the point 2 n for all n ∈ Z.
Applying in turn (3.4), (3.5), we have Vol. 91 (2017) Beta-type functions and the harmonic mean 1047 The validity of is obvious since, by (3.4), f is defined on the interval 2 n , 2 n+1 and the composition and multiplication of continuous functions are continuous. This finishes the proof. Krull's theorem [2,3] (see also [4], pp. 114-115) gives us the existence of a (unique up to a constant) convex solution h to (3.12). Since, for any real constant k, the function u −→ e −u + k satisfies the functional equation (3.12) and is convex, it follows that for some k, h (u) = e −u + k, u ∈ R.