On the Jensen convex and Jensen concave envelopes of means

In recent papers the convexity of quasiarithmetic means was characterized under twice differentiability assumptions. One of the main goals of this paper is to show that the convexity or concavity of a quasiarithmetic mean implies the the twice continuous differentiability of its generator. As a consequence of this result, we can characterize those quasiarithmetic means which admit a lower convex and upper concave quasiarithmetic envelop.


Introduction
for every matrix x ∈ I n×m . Whenever M, N : ∞ n=1 I n → I are both symmetric and repetition-invariant means such that (M, N) is an Ingham-Jessen pair (for all m, n ∈ N) then we can derive several interesting inequalities, among others the mixed-means inequality [2,20] and the Kedlaya inequality [6] (see also [16]) N x 1 , M(x 1 , x 2 ), . . . , M(x 1 , x 2 , . . . , x n ) ≤ M x 1 , N(x 1 , x 2 ), . . . , N(x 1 , x 2 , . . . , x n ) which is valid for all n ∈ N and x ∈ I n ; see also the recent paper [4] for more examples.
In the simplest case when one of the means is the arithmetic mean (from now on denoted by A), we easily obtain: (i) (M, A) is an Ingham-Jessen pair if and only if M is Jensen concave; (ii) (A, N) is an Ingham-Jessen pair if and only if N is Jensen convex. Let us stress that, due to Bernstein-Doetsch theorem [1], in the family of means Jensen convexity and Jensen convexity coincide with convexity and concavity, respectively.
Following the ideas of convex embeddings (hulls, cones and so on) there arises a natural problem: How can we associate a convex (or concave) mean to a given one? A quite similar and comprehensive study related to the homogeneity axiom has been presented recently by the authors [19].
The rest of this paper is split into two parts -we consider convex and concave envelopes in the abstract setting (section 2) and in the quasiarithmetic setting (section 3).
Let us now recall several elementary facts for the family of quasiarithmetic means. This family was axiomatized in 1930s [5,8,10]. For a continuous and strictly monotone function where n ∈ N and a = (a 1 , . . . , a n ) ∈ I n .
The family of all quasiarithmetic means on I will be denoted by Q(I). It was Knopp [7] who noticed that for I = R + and π p (x) := x p (p = 0) and π 0 (x) := ln x, the quasiarithmetic mean A πp coincides with the p-th power mean P p . An important subclass of this family (which contains power means) consists of the means which are generated by C 2 functions with a nowhere vanishing first derivativethis family of generating functions is denoted by C 2# or, more frequently, C 2# (I) if it is necessary to emphasize the domain. Indeed, in view of Jensen's inequality one can easily show that for f, g ∈ C 2# the comparison inequality A f ≤ A g is equivalent to the inequality f ′′ f ′ ≤ g ′′ g ′ which is the comparability of two single-variable functions. For a detailed discussion concerning this relationship, we refer the reader to the paper by Mikusiński [9]. The operator f → f ′′ f ′ was used in several contexts by Pasteczka [11][12][13][14]. There are few approaches to convexity (or concavity) within this family. First, in late 1980s Páles [15] characterized the convexity of so-called quasideviation means (this family contains quasiarithmetic means). Later, the convexity of quasiarithmetic means was characterized by the authors [18] under the assumption that the mean is generated by a function from C 2# . The purpose of this paper is to prove that whenever a quasiarithmetic mean is convex (or concave) then its generator must belong to the class C 2# -see Theorem 3.1 below. An extensive discussion concerning convexity and concavity in the weighted setting has been given recently by Chudziak et al. [3].

Abstract approach to envelopes
In this section we prove a few preliminary results concerning convex and concave envelopes of means. Let us first introduce a formal definition of these operators. Let us emphasize a few simple however important remarks. First, conv S (f ) (resp. conc S (f )) is either finite everywhere or conv S (f ) ≡ −∞ (resp. conc S (f ) ≡ +∞). Second, conv S (f ) is a Jensen convex function (unless conv S (f ) ≡ −∞) and conc S (f ) is a Jensen concave function (unless conc S (f ) ≡ +∞). Third, these operators are monotone function of both f (with the pointwise ordering) and S (with the inclusion ordering). Forth, for every f and S like above the inequality conv S (f ) ≤ f ≤ conc S (f ) holds. Fifth, a function f ∈ S is Jensen convex (resp. Jensen concave) if and only if conv S (f ) = f (resp. conc S (f ) = f ).
2.1. Evelopes in a family of means. Throughout this section let n ∈ N and let I ⊂ R be an interval. An n-variable mean on I is a function M : I n → I such that Family of all such functions will be denoted by M n (I). From now on A n stands for the n-variable arithmetic mean (on I).
We say that a family S ⊆ M n (I) is permutation-closed if, for every M ∈ S and every permutation σ of {1, . . . , n}, the mean M σ : I n → I given by = N x 1 + · · · + x n n , . . . , This implies M(x) ≥ A n (x). As x was taken arbitrarily, we have M ≥ A n . Conversely, if M ≥ A n then, as A n is Jensen convex, we have M ≥ conv S (M) ≥ A n . The proof of the second (i.e. concave) counterpart of this theorem is analogous.

Quasiarithmetic means
In the next result we establish a complete characterization of the convexity of quasiarithmetic means. (1) f ∈ C 2# (I); (2) either f ′′ is nowhere vanishing and f ′ f ′′ is positive and concave, or f ′′ ≡ 0. Proof. The implication (⇐) was already proved by the authors in [18]. It was also proved that, under the assumption that A f is convex, (1) implies (2). Therefore the only remaining part is to show that every convex quasiarithmetic mean is generated by a C 2# function.
As A f is convex, by Theorem 2.2, we obtain A f ≥ A. Thus, by Jensen's inequality and the well-known identity A f = A −f one may assume without loss of generality that f is strictly increasing and convex. Then f has strictly positive one-sided derivatives f ′ + and f ′ − at every point of I.
Applying some general results concerning quasideviation means (cf. [15]), we can obtain that A f is convex if and only if there exist a, b : I 2 → R such that is valid for all x, y, u, v ∈ I. For x > u and y = v we obtain Upon taking the limit x ց u, it follows that Analogously, we obtain , which implies the double inequality .
Take p ∈ I arbitrarily. As f is convex, we know that f ′ − (p) ≤ f ′ + (p) and f is differentiable everywhere except at countably many points. In particular one can take u p ∈ I such that f is differentiable at u p and v p := 2p − u p ∈ I. Then (3.2) with (u, v) : . Consequently, f is differentiable at p. As f is convex, we get f ∈ C 1# (I) (in particular f ′ , is positive). Then, in view of (3.2) and the similar inequality for the function b, one gets Now condition (3.1) can be equivalently rewritten as Which implies that the two-variable continuous function F : I 2 → R given by is convex on I 2 . In particular, for all fixed x ∈ I, the mapping u → F x (u) := F (x, u) is convex on I. Consequently, F x is differentiable at every point of I from the left and from the right. However, as f ∈ C 1# (I), the mapping u → f (x) − f (u) is differentiable.
If U = P , then obviously A h = A for all h ∈ U and conv Q(I) (A f ) = A. From now on assume that the set U 0 := U \ P is nonempty. Then for every h ∈ U 0 we have A h ≥ A and A h = A. In particular, By virtue of Theorem 3.1, for all h ∈ U 0 , we have h ∈ C 2# (I), h ′′ is nowhere vanishing and h ′′ h ′ is positive and concave. Moreover, applying well-known comparability criterion, In particular f ′′ f ′ is positive on its domain. On the other hand, it is relatively easy to verify each function h : I → R satisfying all properties above belongs to U 0 . Therefore By m ≥ f ′ f ′′ we know that m is positive. Thus the 2nd-order linear ordinary differential equation g ′ g ′′ = m has a solution g ∈ C 2# (I). Obviously g ′′ is nowhere vanishing and g ′ g ′′ is positive and concave. Thus Theorem 3.1 implies that A g is convex.
On the other hand, by the definition of the concave envelope for every h ∈ U 0 , we have Applying this inequality to all h ∈ U 0 in view of (3.4) one gets conv Q(I) (A f ) ≤ A g .
To verify the converse inequality, observe that g ′′ g ′ = 1 m ≤ f ′′ f ′ which implies A g ≤ A f . Thus A g is a convex minorant of A f , equivalently g ∈ U 0 . Applying the inequality (3.4) we obtain conv Q(I) (A f ) ≥ A g .
Using Lemma 2.4 we can formulate the result concerning concave envelopes in a family of quasiarithmetic means. Corollary 3.3. Let I be an interval, f ∈ C 2# (I) be an increasing and concave function. Then conc Q(I) (A f ) = A g for some g ∈ C 2# (I).
Moreover either g ′′ ≡ 0 (and A g is the arithmetic mean) or g ′′ is nowhere vanishing and g ′ g ′′ = conv C(I)