Quasi-arithmetic means contain many other mean value concepts
such as the arithmetic, the geometric or the harmonic mean as
special cases. Treating quasi-arithmetic means as sequences of mappings
from I^n into I (for some real interval I) this paper shows that under
mild additional conditions this mapping is uniquely determined by its
values on I^2. This extends a well-known result by Huntington [4] where
this claim is proven only for special cases

Hansen, Maurice

Eldorado - Repository of the TU Dortmund

SFB 823 On axiomizing and extending the quasi-arithmetic mean Discussion Paper  Maurice Hansen     Nr. 12/2018          On axiomizing and extending thequasi-arithmetic meanMaurice HansenAbstract. Quasi-arithmetic means contain many other mean value con-cepts such as the arithmetic, the geometric or the harmonic mean asspecial cases. Treating quasi-arithmetic means as sequences of mappingsfrom In into I (for some real interval I) this paper shows that undermild additional conditions this mapping is uniquely determined by itsvalues on I2. This extends a well-known result by Huntington [4] wherethis claim is proven only for special cases.Mathematics Subject Classiﬁcation (2010). Primary 26E60; Secondary39B22.Keywords. Quasi-arithmetic mean, strong decomposability.The following symbols are used in this paper:N the set of all natural numbers: {1, 2, 3, . . .}N[m,n] {r ∈ N | m ≤ r ≤ n}xT the transpose of x1n the vector (1, . . . , 1)T ∈ Rne(n)i the unit vector (a1, . . . , an)T ∈ Rn with aj ={1 if j = i0 if j = iIf not stated otherwise, the n components of a vector x ∈ Rn are denoted byx1, . . . , xn, such that x = (x1, . . . , xn)T .1. Axioms for meansThe axiomatization of mean value functions has a long and distinguishedhistory (Cauchy [3], Huntington [4], Kolmogoroﬀ [5], Nagumo [7], Marichal[6], among many others); see also Bullen [2, chapter VI, section 6] for a morerecent overview. In compliance with this literature, mean values are deﬁnedas a sequence (Mn)n∈N of mappings Mn : In → I where I is some interval onResearch supported by DFG-Sonderforschungsbereich 823. I am grateful to Walter Kra¨merfor helpful discussions and comments.2 Maurice Hansenthe real line containing more than one element (usually, I ∈ {R,R≥0,R>0}).To qualify as a mean, these functions have to satisfy certain axioms. Foremostamong these is the following requirement:(M1) min{x1, . . . , xn} ≤ Mn(x) ≤ max{x1, . . . , xn} for all n ∈ N, x ∈ In.This condition immediately suggests itself; it was already considered by Cau-chy [3, p. 14] as fundamental for any mean value concept. Additional condi-tions, most of them quite obvious, are:(M2) Mn(c1n) = c for all n ∈ N, c ∈ I (idempotency).(M3) Mn(x) ≤ Mn(y) for all n ∈ N and x, y ∈ In with xi ≤ yi for alli ∈ N[1,n] (isotonicity).(M4) Mn((x1, . . . , xn)T)= Mn((xπ(1), . . . , xπ(n))T)for all n ∈ N, x ∈ Inand all permutations π : N[1,n] → N[1,n] (symmetry).It is easily checked that axioms (M2), (M3) and (M4) are independent,whereas (M1) is implied by (M2) and (M3). Furthermore, the combinationof (M1) and (M3) and the combination of (M2) and (M3) are equivalent.All conventional means obey all these requirements. Additional axiomsare:(M5) Mn : In → I is continuous for all n ∈ N (continuity).(M6) Mn(λx) = λMn(x) for all n ∈ N, x ∈ In, λ ∈ R with λx ∈ In(homogeneity).(M7) Mn(x+ y) = Mn(x) +Mn(y) for all n ∈ N, x, y ∈ In, if a+ b ∈ Iholds for all a, b ∈ I (additivity).(M3′) Mn(x) < Mn(y) for all n ∈ N and x, y ∈ In with xi ≤ yi for alli ∈ N[1,n] and x = y (strict isotonicity).Homogeneity and additivity will play no further role below. Of primaryimportance, however, will be an axiom which can be viewed as a generalisa-tion of what is usually called associativity:(M8) Let n ∈ N, k ∈ N[1,n], x ∈ In and M := Mk((x1, . . . , xk)T). ThenMn((x1, . . . , xk, xk+1, . . . , xn)T)= Mn((M, . . . ,M, xk+1, . . . , xn)T).I. e. substituting a leading subvector of x componentwise by its mean leavesthe overall mean unchanged. This condition has ﬁrst been introduced byBemporad [1] in a characterization of the arithmetic mean. Marichal [6] pointsout that (M8) only makes sense in conjunction with (M4) and suggests ageneralisation which he calls strong decomposability:(M9) For all n ∈ N, k ∈ N[1,n], x ∈ In and any subsetK = {i1, . . . , ik} ⊆ N[1,n] with i1 < i2 < · · · < ik we haveMn(x) = Mn⎛⎝∑i∈KMe(n)i +∑i∈N[1,n]\Kxie(n)i⎞⎠ with M := Mk⎛⎜⎝⎛⎜⎝xi1...xik⎞⎟⎠⎞⎟⎠ .I. e. for any subvector y of x (not necessarily leading or connected), Mnremains unchanged if all elements of y are replaced by Mk(y).Axiomizing the quasi-arithmetic mean 32. Characterizing quasi-arithmetic meansDeﬁnition 2.1. A quasi-arithmetic mean is a sequence (Q[ϕ]n )n∈N of mappingsQ[ϕ]n : In → I withQ[ϕ]n (x) = ϕ−1(1nn∑i=1ϕ(xi))for all n ∈ N, x ∈ In, where ϕ : I → J is a continuous and bijective mapping(see Bullen [2, p. 266]).There are various characterizations of quasi-arithmetic means. Marichal[6] for instance shows that a sequence (Mn)n∈N is a quasi-arithmetic mean ifand only if it obeys axioms (M2), (M3′), (M5) and (M9).Below, Iˆ ⊆ R is an unbounded interval. The following preliminary resultis needed for the central theorem of this paper:Lemma 2.2. Let ϕ : Iˆ → J be continuous and bijective. Then for all n ∈N\{1} and x ∈ Iˆn there exists an a ∈ Iˆ with the following property: For allk ∈ N[1,n−1] one has⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a)), xk+2, . . . , xn⎞⎠T ∈ Iˆn.Proof. We distinguish the following cases:1. Iˆ is neither left-bounded nor right-bounded.2. Iˆ is left-bounded, but not right-bounded.3. Iˆ is right-bounded, but not left-bounded.In each case, one has to show that all components of the vector belong to Iˆ.1. If Iˆ is neither left-bounded nor right-bounded, then Iˆ = R and thestatement holds for any a ∈ R.2. Let n ∈ N, x ∈ Iˆn, choose a := min{x1, . . . , xn}, let k ∈ N[1,n−1]. Byassumption a, xk+2, . . . , xn ∈ Iˆ. Show ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))∈ Iˆ.If ϕ is strictly increasing, the following holds:a = min{x1, . . . , xn} ⇒ a ≤ xi for all i ∈ N[1,k+1]⇒ ϕ(a) ≤ ϕ(xi) for all i ∈ N[1,k+1]⇒ (k + 1)ϕ(a) ≤k+1∑i=1ϕ(xi)⇒ ϕ(a) ≤k+1∑i=1ϕ(xi)− kϕ(a)⇒ a ≤ ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a))4 Maurice HansenHence ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))∈ Iˆ, since a ∈ Iˆ and Iˆ is not right-bounded.If ϕ is strictly decreasing, one obtains the conclusion in the sameway:a = min{x1, . . . , xn} ⇒ a ≤ xi for all i ∈ N[1,k+1]⇒ ϕ(a) ≥ ϕ(xi) for all i ∈ N[1,k+1]⇒ (k + 1)ϕ(a) ≥k+1∑i=1ϕ(xi)⇒ ϕ(a) ≥k+1∑i=1ϕ(xi)− kϕ(a)⇒ a ≤ ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a))Again ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))∈ Iˆ, since a ∈ Iˆ and Iˆ is not right-bounded.3. Let n ∈ N, x ∈ Iˆn, choose a := max{x1, . . . , xn}, let k ∈ N[1,n−1]. Byassumption a, xk+2, . . . , xn ∈ Iˆ. Show ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))∈ Iˆ.If ϕ is strictily increasing, the following holds:a = max{x1, . . . , xn} ⇒ a ≥ xi for all i ∈ N[1,k+1]⇒ ϕ(a) ≥ ϕ(xi) for all i ∈ N[1,k+1]⇒ (k + 1)ϕ(a) ≥k+1∑i=1ϕ(xi)⇒ ϕ(a) ≥k+1∑i=1ϕ(xi)− kϕ(a)⇒ a ≥ ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a))Hence ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))∈ Iˆ, since a ∈ Iˆ and Iˆ is not left-bounded.If ϕ is strictly decreasing, one obtains the conclusion in the sameway. The following theorem shows that a quasi-arithmetic mean is in a senseuniquely determined by its values on Iˆ2:Theorem 2.3. Let (Mn)n∈N be a sequence of mappings Mn : Iˆn → Iˆ. If(Mn)n∈N satisﬁes (M2), (M9) and M2 = Q[ϕ]2 for a continuous and bijectivemapping ϕ : Iˆ → J , then Mn = Q[ϕ]n for all n ∈ N.Axiomizing the quasi-arithmetic mean 5Proof. Consider n = 1. Because of (M2) the following holds for all x ∈ Iˆ:M1(x) = M1(x11) = x = Q[ϕ]1 (x)For n ∈ N\{1}, we ﬁrst show that for all x ∈ Iˆn one hasMn(x) = Mn⎛⎜⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a)), xk+2, . . . , xn⎞⎠T⎞⎟⎠for all k ∈ N[1,n−1], if a is chosen according to lemma 2.2.Let n ∈ N\{1}, x ∈ Iˆn and choose a according to lemma 2.2. First ofall, the termMn⎛⎜⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a)), xk+2, . . . , xn⎞⎠T⎞⎟⎠is well-deﬁned according to lemma 2.2 for all k ∈ N[1,n−1]. The claim is shownby induction over k ∈ N[1,n−1].Base case: If k = 1, one hasMn(x) = Mn((x1, . . . , xn)T)(M9)= Mn((M2((x1x2)),M2((x1x2)), x3, . . . , xn)T)= Mn((Q[ϕ]2((x1x2)),Q[ϕ]2((x1x2)), x3, . . . , xn)T). (1)FurthermoreM2((aϕ−1(ϕ(x1) + ϕ(x2)− ϕ(a))))= Q[ϕ]2((aϕ−1(ϕ(x1) + ϕ(x2)− ϕ(a))))= ϕ−1(ϕ(a) + ϕ(ϕ−1(ϕ(x1) + ϕ(x2)− ϕ(a)))2)= ϕ−1(ϕ(x1) + ϕ(x2)2)= Q[ϕ]2((x1x2)),henceMn(x)(1)= Mn((Q[ϕ]2((x1x2)),Q[ϕ]2((x1x2)), x3, . . . , xn)T)(M9)= Mn((a, ϕ−1(ϕ(x1) + ϕ(x2)− ϕ(a)), x3, . . . , xn)T).6 Maurice HansenInductive step: SupposeMn(x) = Mn⎛⎜⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a)), xk+2, . . . , xn⎞⎠T⎞⎟⎠ .holds for k ∈ N[1,n−2]. Since k ≤ n − 2, ϕ−1(∑k+1i=1 −kϕ(a))is not the lastcomponent of the vector, so xk+2 exists. One hasMn(x)= Mn⎛⎜⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a)), xk+2, . . . , xn⎞⎠T⎞⎟⎠(M9)= Mn⎛⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times,M2((ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))xk+2)),M2((ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))xk+2)), xk+3, . . . , xn)T⎞⎠= Mn⎛⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times,Q[ϕ]2((ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))xk+2)),Q[ϕ]2((ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a))xk+2)), xk+3, . . . , xn)T⎞⎠= Mn⎛⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1⎛⎝ϕ(ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a)))+ ϕ(xk+2)2⎞⎠ ,ϕ−1⎛⎝ϕ(ϕ−1(∑k+1i=1 ϕ(xi)− kϕ(a)))+ ϕ(xk+2)2⎞⎠ ,xk+3, . . . , xn)T)= Mn⎛⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1(∑k+2i=1 ϕ(xi)− kϕ(a)2),ϕ−1(∑k+2i=1 ϕ(xi)− kϕ(a)2), xk+3, . . . , xn)T⎞⎠ . (2)(Clearly, the components xk+3, . . . , xn at the end of the vectors are omittedif k = n− 2.)Axiomizing the quasi-arithmetic mean 7FurthermoreM2((aϕ−1(∑k+2i=1 ϕ(xi)− (k + 1)ϕ(a))))= Q[ϕ]2((aϕ−1(∑k+2i=1 ϕ(xi)− (k + 1)ϕ(a))))= ϕ−1⎛⎝ϕ(a) + ϕ(ϕ−1(∑k+2i=1 ϕ(xi)− (k + 1)ϕ(a)))2⎞⎠= ϕ−1(∑k+2i=1 ϕ(xi)− kϕ(a)2),henceMn(x)(2)= Mn⎛⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1(∑k+2i=1 ϕ(xi)− kϕ(a)2),ϕ−1(∑k+2i=1 ϕ(xi)− kϕ(a)2), xk+3, . . . , xn)T⎞⎠(M9)= Mn⎛⎜⎜⎝⎛⎜⎝ a, . . . , a︸ ︷︷ ︸(k+1)-times, ϕ−1(k+2∑i=1ϕ(xi)− (k + 1)ϕ(a)), xk+3, . . . , xn⎞⎟⎠T⎞⎟⎟⎠ .This shows thatMn(x) = Mn⎛⎜⎝⎛⎝a, . . . , a︸ ︷︷ ︸k-times, ϕ−1(k+1∑i=1ϕ(xi)− kϕ(a)), xk+2, . . . , xn⎞⎠T⎞⎟⎠holds for all k ∈ N[1,n−1] and n ∈ N\{1}, x ∈ Iˆn.In particular, setting k = n− 1, one hasMn(x) = Mn⎛⎜⎜⎝⎛⎜⎝ a, . . . , a︸ ︷︷ ︸(n−1)-times, ϕ−1(n∑i=1ϕ(xi)− (n− 1)ϕ(a))⎞⎟⎠T⎞⎟⎟⎠ . (3)8 Maurice HansenThis implies thatMn(x)(3)= Mn⎛⎜⎜⎝⎛⎜⎝ a, . . . , a︸ ︷︷ ︸(n−1)-times, ϕ−1(n∑i=1ϕ(xi)− (n− 1)ϕ(a))⎞⎟⎠T⎞⎟⎟⎠= Mn⎛⎜⎜⎝⎛⎜⎝ a, . . . , a︸ ︷︷ ︸(n−1)-times, ϕ−1(nϕ(Q[ϕ]n (x))− (n− 1)ϕ(a))⎞⎟⎠T⎞⎟⎟⎠= Mn⎛⎜⎜⎝⎛⎜⎝ a, . . . , a︸ ︷︷ ︸(n−1)-times, ϕ−1(n∑i=1ϕ(Q[ϕ]n (x))− (n− 1)ϕ(a))⎞⎟⎠T⎞⎟⎟⎠ .Moreover, one hasMn(Q[ϕ]n (x)1n)(3)= Mn⎛⎜⎜⎝⎛⎜⎝ a, . . . , a︸ ︷︷ ︸(n−1)-times, ϕ−1(n∑i=1ϕ(Q[ϕ]n (x))− (n− 1)ϕ(a))⎞⎟⎠T⎞⎟⎟⎠ ,soMn(x) = Mn(Q[ϕ]n (x)1n)(M2)= Q[ϕ]n (x).Hence, Mn = Q[ϕ]n also holds for all n ∈ N\{1}. Theorem 2.3 easily allows for a characterization of speciﬁc quasi-arith-metic means by independent axioms, as shown by Huntington [4] for thearithmetic, geometric and harmonic mean and the root mean square. AsHuntington operated without the strong decomposability axiom, he had torely on a weaker symmetry axiom instead. With strong decomposability, thesymmetry requirement is redundant. The following theorem can therefore beviewed as a generalisation of Huntington’s result:Theorem 2.4.1. Let (Mn)n∈N be a sequence of mappings Mn : Iˆn → Iˆ, let ϕ : Iˆ → Jbe continuous and bijective. Then (Mn)n∈N satisﬁes (M2), (M9) andM2 = Q[ϕ]2 if and only if Mn = Q[ϕ]n for all n ∈ N.2. Let ϕ : I → J be continuous and bijective. Then the three properties(M2), (M9) and M2 = Q[ϕ]2 are independent (i. e none of the axioms isimplied by the others).Proof.1. Necessity: See Theorem 2.3.Suﬃciency: Trivial, since (Q[ϕ]n )n∈N satisﬁes (M2) and (M9).Axiomizing the quasi-arithmetic mean 92. Choose any ﬁxed c ∈ I, then consider the sequences (Mn)n∈N, (Nn)n∈Nand (On)n∈N of mappings Mn, Nn, On : In → I withMn(x) ={Q[ϕ]n (x) , n ∈ {1, 2}c , n ≥ 3Nn(x) ={Q[ϕ]n (x) , n ∈ {1, 2}Mn(x) , n ≥ 3On(x) = x1for all n ∈ N, x ∈ In. Here, Mn(x) denotes the median of x.• (Mn)n∈N satisﬁes (M9): Let n ∈ N, k ∈ N[1,n], x ∈ In and K ={i1, . . . , ik} ⊆ N[1,n] with i1 < i2 < · · · < ik. If n ∈ {1, 2}, it isclear thatMn(x) = Mn⎛⎜⎝∑i∈KMk⎛⎜⎝⎛⎜⎝xi1...xik⎞⎟⎠⎞⎟⎠ e(n)i + ∑i∈N[1,n]\Kxie(n)i⎞⎟⎠holds, since Mn(x) = Q[ϕ]n (x) for all x ∈ In in this case and thequasi-arithmetic mean is strictly decomposable. If n ≥ 3, thenMn(x) = c andMn⎛⎜⎝∑i∈KMk⎛⎜⎝⎛⎜⎝xi1...xik⎞⎟⎠⎞⎟⎠ e(n)i + ∑i∈N[1,n]\Kxie(n)i⎞⎟⎠ = cby deﬁnition, thus (Mn)n∈N satisﬁes (M9). Furthermore M2 =Q[ϕ]2 . However, property (M2) is not satisﬁed: Choose d ∈ I withd = c, thenM3(d13) = c = d.• (Nn)n∈N statisﬁes (M2) and N2 = Q[ϕ]2 . However, property (M9)is not satisﬁed: Choose a, b, c, d, e ∈ I with a < b < c < d < e,thenN5((a, b, c, d, e)T)= c = N5((b, b, b, d, e)T)= b.• (On)n∈N satisﬁes (M2), which is clear, and also (M9): Let n ∈ N,k ∈ N[1,n], x ∈ In and K = {i1, . . . , ik} ⊆ N[1,n] with i1 < i2 <10 Maurice Hansen· · · < ik. ThenOn⎛⎜⎝∑i∈KOk⎛⎜⎝⎛⎜⎝xi1...xik⎞⎟⎠⎞⎟⎠ e(n)i + ∑i∈N[1,n]\Kxie(n)i⎞⎟⎠=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Ok⎛⎜⎜⎝⎛⎜⎜⎝xi1...xik⎞⎟⎟⎠⎞⎟⎟⎠ , i1 = 1x1 , i1 = 1={xi1 , i1 = 1x1 , i1 = 1= x1= On(x).However, there is no continuous and bijective mapping ϕ : I → Jsuch thatϕ−1(ϕ(x1) + ϕ(x2)2)= x1holds for all x1, x2 ∈ I. Since ϕ is a bijection, the following holdsfor all x1, x2 ∈ I with x1 = x2:ϕ(x1) = ϕ(x2) ⇒ ϕ(x1) + ϕ(x2)2= ϕ(x1)⇒ ϕ−1(ϕ(x1) + ϕ(x2)2)= x1. References[1] Bemporad, G.: Sul principio della media aritmetica. Atti Accad. Naz. Lincei6(3), 87–91 (1926)[2] Bullen, P. S.: Handbook of Means and Their Inequalities. Kluwer, Dordrecht(2003)[3] Cauchy, A. L.: Cours d’analyse de l’Ecole Royale Polytechnique. Vol. I. Analysealge´brique. Debure, Paris (1821)[4] Huntington, E. V.: Sets of Independent Postulates for the Arithmetic Mean,the Geometric Mean, the Harmonic Mean and the Root-Mean-Square. Trans.Amer. Math. Soc. 29(1), 1–22 (1927)[5] Kolmogoroﬀ, A. N.: Sur la notion de la moyenne. Atti Accad. Naz. Lincei Mem.Cl. Sci. Fis. Mat. Natur. Sez 6(12), 388–391 (1930)[6] Marichal, J.-L.: On an axiomatization of the quasi-arithmetic mean values with-out the symmetry axiom. Aequationes Math. 59(1), 74–83 (2000)[7] Nagumo, M.: U¨ber eine Klasse der Mittelwerte. Japan J. Math. 7, 71–79 (1930)Axiomizing the quasi-arithmetic mean 11Maurice HansenTechnische Universita¨t DortmundFakulta¨t StatistikCDI-Geba¨ude44221 DortmundGermanye-mail: maurice.hansen@tu-dortmund.de   

On axiomizing and extending the quasi-arithmetic mean

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On axiomizing and extending the quasi-arithmetic mean

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