Generalization of the Harmonic Weighted Mean Via Pythagorean Invariance Identity and Application

Abstract Under some simple conditions on the real functions f and g defined on an interval I ⊂ (0, ∞), the two-place functions Af (x, y) = f (x) + y − f (y) and Gg(x,y)=g(x)g(y)y {G_g}\left({x,y} \right) = {{g\left(x \right)} \over {g\left(y \right)}}y generalize, respectively, A and G, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity G ∘ (H, A) = G (equivalent to the Pythagorean harmony proportion), a suitable weighted extension Hf,g of the classical harmonic mean H is introduced. An open problem concerning the symmetry of Hf,g is proposed. As an application a method of effective solving of some functional equations involving means is presented.


Introduction
Let I ⊂ R be an interval and let f, ϕ : I → R be arbitrary functions. In [8] it was shown that the two-variable function M : I 2 → R of the form M (x, y) = f (x) + ϕ (y) , x, y ∈ I, is a mean in I, i.e. that min (x, y) ≤ M (x, y) ≤ max (x, y) , x, y ∈ I, if, and only if, ϕ = id | I −f , and the functions f and id | I −f are non-decreasing, and the function M = A f defined by A f (x, y) = f (x) + y − f (y) , strictly related to addition, generalizes the weighted arithmetic mean. In particular, if A f is symmetric then A f = A, where A (x, y) := x+y 2 (see Proposition 1). Let I ⊂ (0, ∞) and let g, ψ : I → (0, ∞) . In the present note we first observe that the function M : I 2 → R of the form M (x, y) = g (x) ψ (y) , x, y ∈ I, is a mean in I, iff ψ = id | I g and both functions g, id | I g are non-decreasing.
Moreover the mean M = G g defined by G g (x, y) = g(x) g(y) y, strictly related to multiplication, generalizes the weighted geometric mean. In particular, if G g is symmetric then G g = G, where G (x, y) = √ xy (see Proposition 2).
The generalizations of the weighted arithmetic and geometric means given by Proposition 1 and Proposition 2, are based, respectively, on relation of the classical arithmetic mean A to addition, and the geometric mean G to multiplication.
Having these generalizations, a legitimate question arises if the classical harmonic mean H (x, y) = 2xy x+y can be also extended. The main result of the present paper, Theorem 1 in section 4, gives the affirmative answer. It turns out that, with the aid of the weighted extensions A f and G g of the arithmetic and geometric means, basing on the identity G • (H, A) = G, the invariance of the geometric mean G with respect to the mean-type mapping (H, A) (equivalent to the classical Pythagorean harmony proportion), one can obtain the means H f,g generalizing the harmonic mean. In section 5 the symmetry of the mean H f,g is considered and an open problem is proposed. In section 6, given H f,g , we ask for its harmonically complementary mean H ϕ,ψ , i.e. such that H •(H f,g , H ϕ,ψ ) = H. In section 7 we apply Theorem 1 to obtain the effective form of the continuous solutions Φ of functional equations of form Φ (H f,g (x, y) , A f (x, y)) = Φ (x, y) .

Generalization of weighted arithmetic mean
We begin with recalling the following Proposition 1 (see [8]). Let I ⊂ R be an interval and let f, ϕ : I → R. Then the function and the functions f and id | I − f are non-decreasing. Moreover (i) A f is a mean iff the function f is non-decreasing and non-expansive; (ii) A f is a strict mean iff f and id | I − f are strictly increasing, or equivalently, iff f is strictly increasing and strictly contractive; Remark 1. Let a, b, c be positive real numbers and let f : (0, ∞) → R be defined by Then Proof. It is enough to note that, for all x > 0, are positive, so the assumptions of Proposition 1 are satisfied.
Let us also note some general properties of functions of the form A f , not assuming that A f is a mean.

Generalization of weighted geometric mean
It is easy to prove the following Proposition 2. Let I ⊂ (0, ∞) be an interval and let g, ψ : I → (0, ∞). Then the function y, x, y ∈ I, and the functions g and id | I g are non-decreasing. Moreover (i) G g is a mean iff the function g is non-decreasing and ≤ y x , x, y ∈ I, x < y; (ii) G g is a strict mean iff g and id | I g are strictly increasing, or equivalently, iff < y x , x, y ∈ I, x < y; (iii) G g is symmetric iff G g = G, or equivalently, iff the function g(x) √ x is constant in I.
To determine a possible broad class of functions g : (0, ∞) → (0, ∞) , being good candidates for generating the generalized geometric means, let us fix p ∈ (0, 1], write g in the form where γ : (0, ∞) → (0, ∞) should be chosen in such a way that the functions g and id | I g are increasing. Since the function γ should be such that or equivalently, such that The homographic function (0, ∞) x −→ ax+b cx+1 with positive parameters a, b, c > 0 satisfies the inequality Solving the differential equation for some d > 0. Setting q := −c (a − bc) , p = b, r = c, and taking into account that 0 < a ≤ c, we hence get the following and g : (0, ∞) → (0, ∞) is given by then the functions g and id | I g are increasing, and strictly increasing if p ∈ (0, 1) . (Of course, without any loss of generality, one can take d = 1.) Let us also note some general properties of functions of the form G g , not assuming that G g is a mean.
G g (tx, ty) = tG g (x, y) , t, x, y > 0; (iv) the function g is multiplicative, i.e., Indeed, assume that G g is sub-homogeneous. Replacing x by x t and y by x, y, which shows that G g is super-homogeneous, so it is homogeneous. The implication (iii) =⇒ (iv) is obvious. For the "moreover" part see [1], [4].

Main result -generalization of weighted harmonic mean
The above considerations lead to the natural question if one can define a relevant counterpart of the harmonic mean H (x, y) = 2xy x + y , x, y > 0.
In this section we show that, basing on the classical invariance identity G • (H, A) = G, and applying the above generalizations of the arithmetic and geometric means, one can give a positive answer. Namely, we prove the following Theorem 1. Let I ⊂ (0, ∞) be an interval and assume that f : I → R, g : I → (0, ∞) be such that the functions f, id | I − f , g and id | I g are strictly increasing. Then (i) f, g are continuous, and the function H f,g : (ii) if moreover the function y is (strictly) increasing in both variables, then H f,g is a (strict) mean in I; (iv) if (2) or (3) holds, the mean G g is invariant with respect to the meantype mapping (H f,g , A f ) : Proof. (i) The continuity of the functions f and g follows from the assumed increasing monotonicity of the functions f, id | I − f , g and id | I g . For every x ∈ I, by the definition of A f and G g , we have so H f,g is reflexive.
(ii) Take x, y ∈ I. If x < y then, as g is strictly increasing, this inequality is equivalent to Since the function id | I g is increasing and x < A f (x, y) < y we have , .

Multiplying both sides by g (x)
gives whence, by the monotonicity of g and the definition of H f,g , we get Since x < A f (x, y) < y, applying in turn the monotonicity of g and (2) we have Hence, by the monotonicity of g and the definition of H f,g , If x > y the argument is similar, so we omit it. Thus, for all x, y ∈ I, x = y , we have which shows that H f,g is a strict mean.
To prove (iii) take arbitrary x, y ∈ I. Without any loss of generality, we can assume that x < y, so that x = min (x, y) and y = max (x, y). Applying the reflexivity of H f,g , the definitions of H f,g , A f , G g , increasing strict monotonicity of the function (3) with respect to both variables and increasing strict monotonicity of g −1 , we obtain so H f,g is a strict mean. The monotonicity of H f,g is obvious.
(iv) Note that, for all x, y ∈ I, by the definitions of G g , A f and (1), we have so (4) holds, that is G g is invariant with respect to the mean-type mapping (H f,g , A f ).
The remaining statement follows from the main result in [6] (cf. also [7]).
To illustrate an application of this result recall that if a, b, c are positive real numbers such that a ≤ c and b < 1 , then, in view of Remark 1, the function A f with f : (0, ∞) → R given by f (x) = ax 2 +bx cx+1 is a mean in (0, ∞).
Using these functions and Theorem 1 we give two examples of generalizations of the classical harmonic mean.
Example 1. For the functions f and g with a = 1, b = 1 2 , c = 2, an arbitrary p ∈ (0, 1], and q = 0, we have the above function is increasing with respect to both variables if p ≥ 1 2 . By Theorem 1, taking into account that g −1 (x) = x 1/p for all x > 0, we conclude that, for every p ∈ 1 2 , 1 , the function is a mean in (0, ∞). In particular, taking p = 1 2 , we obtain H f,g (x, y) = 2xy x + y , x, y > 0, so, in this case, H f,g coincides with the classical harmonic mean.
Example 2. Taking the functions f and g with a = 1, b = 1 2 , c = 2; p ∈ 1 2 , 1 , and q = p − 1 2 , one can check that the function is increasing with respect to each variable. Consequently, by Theorem 1, the function H f,g is a mean. In particular, in the case p = 1 we obtain and Remark 6. Since, in view of Theorem 1, the mean H f,g is a composition of the bivariable function g −1 (g (u) g (v)) , the mapping u, g −1 v u and the means A f ,G g (compare [3] where the compositions of quasi-arithmetic means are considered; see also [9]).

Symmetry of generalized weighted harmonic mean and an open question
It is easy to verify that the generalized weighted means A f and G g are symmetric iff they coincide with the classical arithmetic geometric means A and G, respectively. It is interesting that the problem of symmetry of the generalized weighted mean H f,g appears to be nontrivial. To show it we begin with the following Proposition 3. Let I ⊂ (0, ∞) be an interval and t ∈ (0, 1). Assume that f (x) := tx for x ∈ I → R, the function g : I → (0, ∞) is increasing, differentiable and the function id | I g is increasing.
The mean H f,g : I 2 → (0, ∞) defined by (1) is symmetric, i.e. Proof. Since A f (x, y) = tx + (1 − t) y is the standard weighted arithmetic mean in I, in view of formula (1), y , x, y ∈ I.
Thus H f,g is symmetric iff x, x, y ∈ I, which is equivalent to Differentiating both sides with respect to x and then setting y = x we get Solving this differential equation we obtain for some c > 0. Now, from the definition of H f,g , after simple calculations, we get x for all x ∈ I, and, consequently, A f (x, y) = x+y 2 = A(x, y); G g (x, y) = √ xy = G(x, y) and, H f,g (x, y) = 2xy x+y = H(x, y) for all x, y ∈ I.
The mean H f,g : and H f,g = H, where A, G, H are the classical symmetric arithmetic, geometric and harmonic means.
Proof. Since G g (x, y) = x t y 1−t is the weighted arithmetic mean in I, making use of (1), the mean H f,g is symmetric iff By the definition of A f , setting here we get Thus f is of the class C ∞ in I and, clearly, for all x, y ∈ I, it follows that p = 0 and, by the definition of p, we get t = 1 2 . Hence g (x) = √ x and f (x) = x 2 + c for some c > 0. It follows that A f = A, G g = G and H f,g = H.
In this context a natural and open question arises: Problem 1. Is it generally true that the generalized harmonic mean H f,g is symmetric if, and only if, H f,g = H, A f = A and G g = G?
The above two propositions seem to suggest that the answer is affirmative. In this case it would strengthen the central position of the classical means in the rich family of means, and the very special role of the Pythagorean harmony proportion identity.
Remark 7. The mean H f,g is symmetric iff x, x, y ∈ I, or, equivalently, iff x, x, y ∈ I.
Assuming that f, g are three times differentiable, taking derivative of both sides in x, and then setting y = x, we get If g (x) − xg (x) = 0 then xg (x) = 0 and, consequently, we would have g (x) = 0, contradicting the assumption that g (x) is positive for every x ∈ I.
It follows that and, obviously, In this connection, taking into account the key role played by the invariant means in effective finding the limits of the iterates of the mean-type mappings, as well as the considerations in the previous section, the following crucial question arises.
Given an interval I ⊂ (0, ∞), determine all functions f, g, ϕ, ψ : I → (0, ∞), satisfying the suitable conditions of Theorem 1, such that i.e. such that the generalized weighted means H f,g and H ϕ,ψ are complementary with respect to the harmonic mean H.
Assume that H f,g is a mean. Since H is continuous, symmetric and increasing in each variable, there exists a unique mean N : I 2 → I such that H • (H f,g , N ) = H (see Remark 1 in [5]). So the question is if there are ϕ and ψ such that N = H ϕ,ψ . Note that this equality is equivalent to the functional equation y)] , x, y ∈ I.

An application
Theorem 2. Let I ⊂ (0, ∞) be an interval. Assume that the functions f : I → R, g : I → (0, ∞) are such that f, id | I − f , g and id | I g are strictly increasing, and the function is increasing with respect to each variable. Then: (i) A function Φ : I 2 → R, continuous on the diagonal {(x, x) : x ∈ I} , satisfies the functional equation where (H f,g , A f ) n denotes the nth iterate of the mean-type mapping (H f,g , A f ) . In view of Theorem 1 we have lim n→∞ (H f,g , A f ) n (x, y) = (G g (x, y) , G g (x, y)) , x, y ∈ I.
To prove the converse implication, assume that there is a function ϕ: I → R such that (7.2) holds. Then, by Theorem 1(iii), the mean G g is invariant with respect to the mean-type mapping (H f,g , A f ) , and, for all x, y ∈ I, Φ (x, y) = ϕ•G g (x, y) = ϕ•(G g (H f,g , A f )) (x, y) = Φ (H f,g (x, y) , A f (x, y)) , which proves that Φ satisfies functional equation (7.1).
(ii) Since every mean is reflexive and continuous on the diagonal, the result follows from (i).