Functional Inequalities for Metric-Preserving Functions with Respect to Intrinsic Metrics of Hyperbolic Type

: We obtain functional inequalities for functions which are metric-preserving with respect to one of the following intrinsic metrics in a canonical plane domain: hyperbolic metric or some restrictions of the triangular ratio metric, respectively, of a Barrlund metric. The subadditivity turns out to be an essential property, being possessed by every function that is metric-preserving with respect to the hyperbolic metric and also by the composition with some speciﬁc function of every function that is metric-preserving with respect to some restriction of the triangular ratio metric or of a Barrlund metric. We partially answer an open question, proving that the hyperbolic arctangent is metric-preserving with respect to the restrictions of the triangular ratio metric on the unit disk to radial segments and to circles centered at origin.


Introduction
Metric-preserving functions have been studied in general topology from a theoretical point of view and have applications in fixed point theory [1,2], as well as in metric geometry to construct new metrics from known metrics, as the metrics d 1+d , log(1 + d) and the α−snowlake d α , α ∈ (0, 1) associated to each metric d [3][4][5]. The theory of metric-preserving functions, that can be traced back to Wilson and Blumenthal, has been developed by Borsík, Doboš, Piotrowski, Vallin [6][7][8][9] and others, being recently generalized to semimetric spaces and quasimetric spaces [10] (see also [11][12][13]). As we will show below, there is a strong connection between metric-preserving functions and subadditive functions. The theory of subadditive function is well-developed [14,15], the functional inequality corresponding to subadditivity being viewed as a natural counterpart of Cauchy functional equation [16,17].
Given a function f : [0, ∞) → [0, ∞), it is said that f is metric-preserving if for every metric space (X, d) the function f • d is also a metric on X, i.e., f transfers every metric to a metric The function f : If there exists some metric space (X, d) such that the function f • d is also a metric on X, then f : [0, ∞) → [0, ∞) is amenable. The symmetry axiom of a metric is obviously satisfied by f • d whenever d is a metric. Given f amenable, f is metric-preserving if and only if f • d satisfies triangle inequality whenever d is a metric.
Each of the following properties is known to be a sufficient condition for an amenable function to be metric-preserving [10,11]: 1.
f is tightly bounded (that is, there exists a > 0 such that f (x) ∈ [a, 2a] for every x > 0).
We are interested in the following problem: given a particular metric d on a subset A of the complex plane, find necessary conditions satisfied by amenable functions f : [0, ∞) → [0, ∞) for which f • d is a metric. In other terms, we look for solutions of the functional inequality f (d(x, z)) ≤ f (d(x, y)) + f (d(y, z)) for all x, y, z ∈ A.
If we can find for every a, b ∈ [0, ∞) some points x, y, z ∈ A such that d(x, y) = a, d(y, z) = b and d(x, z) = a + b, then f is subadditive on [0, ∞). For some metrics d it could be difficult or impossible to find such points.
We will consider the cases where d is a hyperbolic metric, a triangular ratio metric or some other Barrlund metric. Recall that all these metrics belong to the class of intrinsic metrics, which is recurrent in the study of quasiconformal mappings [4].
The hyperbolic metric ρ D on the unit disk D is given by that is, ρ D (x, y) = 2arctanhp D (x, y), where p D (x, y) = |x−y| |1−xy| is the pseudo-hyperbolic distance and we denoted by arctanh the inverse of the hyperbolic tangent tanh [19]. The hyperbolic metric ρ H on the upper half plane H is given by For every simply-connected proper subdomain Ω of C one defines, via Riemann mapping theorem, the hyperbolic metric ρ Ω on Ω. We prove that, given f : [0, ∞) → [0, ∞), if f • ρ Ω is a metric on Ω, then f is subadditive. In the other direction, if f : [0, ∞) → [0, ∞) is amenable, nondecreasing and subadditive, then f • ρ Ω is a metric on Ω.
The triangular ratio metric s G of a given proper subdomain G ⊂ C is defined as follows for x, y ∈ G [20] For the triangular ratio metric s H on the half-plane, it is known that s H (x, y) = tanh The triangular ratio metric s D (x, y) on the unit disk can be computed analytically as s D (x, y) = |x−y| |x−z 0 |+|z 0 −y| , where z 0 ∈ ∂D is the root of the algebraic equation for which |x − z| + |z − y| has the least value [21]. However, a simple explicit formula for s D (x, y) is not available in general.
As arctanhs H is a metric on the upper half-plane H, it is natural to ask if arctanhs D is a metric on the unit disk D. The answer is unknown, but we prove that some restrictions of arctanhs D are metrics, namely the restriction to each radial segment of the unit disk and the restriction to each circle |z| = ρ < 1. Given f : [0, 1) → [0, ∞) such that the restriction of f • s D to some radial segment of the unit disk D is a metric, we prove that f • tanh is subadditive on [0, ∞). If a continuous amenable function F : [0, 1) → [0, ∞) is metric-preserving with respect to the restriction of the triangular ratio metric s D to every circle |z| = r < 1, r ∈ (0, 1), we prove that for every a, b ∈ [0, ∞) we have For a proper subdomain G ⊂ R n , for a number p ≥ 1 , and for points x, y ∈ G , let The above formula defines a metric, as shown by A. Barrlund [22] for G = R n \ {0} and by P. Hästö [5] in the general case. This metric is called a Barrlund metric and is studied in [23]. In addition, the limit case p = ∞ is considered and it is shown that the formula , x, y ∈ G defines a metric. Note that b G,p is invariant to similarities for every p ∈ [1, ∞] and that for p = 1 the Barrlund metric coincides with the triangular ratio metric. We will consider Barrlund metrics with p = 2 on canonical domains in plane, the upper half plane and the unit disk, that have explicit formulas [23]. For G ∈ {H, D}, assuming that F • b G,2 is a metric on some subset A ⊂ G which is a ray, a line or a circle, we obtain a functional inequality satisfied by F, under the form of the subadditivity of a composition F • ϕ, where the function ϕ depends only on A.

The Case of Hyperbolic Metrics
Let D be the unit disk with the hyperbolic metric ρ D . Let f : [0, ∞) → [0, ∞) be amenable. If f is subadditive and nondecreasing, then f • ρ D is a metric on the unit disk D.
On the other hand, Conversely, if (4) holds and g is nondecreasing, then (3) is satisfied.
Denoting ρ = ϕ(r) and σ = ϕ(s), (4) is equivalent to Now, denoting ln ρ = −u and ln σ = −v, we see that the above condition reduces to We proved that f is subadditive.

Remark 1.
The functional equation associated to the inequality (4) g r+s 1+rs = g(r) + g(s), r, s ∈ [0, 1) reduces via the substitution h(t) = g tanh t 2 ) to the Cauchy equation h(u + v) = h(u) + h(v), u, v ∈ [0, ∞). Extending h to an odd function, we may assume that h is additive on R. If g is bounded on one side on a set of positive Lebesgue measure, then h is linear [16]; hence, there exists some positive constant c such that g(t) = carctanh(t), t ∈ [0, ∞).
Let H be the upper half-plane with the hyperbolic metric ρ H . We are interested in the amenable functions f : [0, ∞) → [0, ∞) for which f • ρ H is a metric on H. Consider the Cayley transform T : H → D, T(z) = z−i z+i , which is a bijective conformal map. Noting that for all x, y ∈ H we have ρ H (x, y) = ρ D (T(x), T(y)), it follows that f • ρ H is a metric on H if and only if f • ρ D is a metric on D. From Proposition 1 we get the following is amenable and f • ρ H is a metric on upper half-plane H, then f is subadditive.
Clearly, f • ρ Ω is a metric on Ω if and only if f • ρ D is a metric on D. Now Proposition 1 leads to following generalization of itself.

The Case of Unbounded Geodesic Metric Spaces
We can give another proof of Theorem 1, based on geometric arguments in geodesic metric spaces. The main idea is that in a geodesic metric space the distance is additive along geodesics.
A topological curve γ : I → X in a metric space (X, d), where I ⊂ R is an interval, is called a geodesic if L γ| [t 1 ,t 2 ] = d(γ(t 1 ), γ(t 2 )) for every subinterval [t 1 , t 2 ] ⊂ I, i.e., the length of every arc of the geodesic is equal to the distance between the endpoints of the arc. A metric space is called a geodesic metric space if every pair of its points can be joined by a geodesic path. Lemma 1. In a geodesic metric space (X, d) that is unbounded, for every positive numbers a and b there exists some points x, y, z ∈ X such that d(x, y) = a, d(y, z) = b and d(x, z) = a + b.
Proof. Let a, b be positive numbers. Fix an arbitrary point x ∈ X. As (X, d) is unbounded, there exists a point w ∈ X such that d(x, w) > a + b. As (X, d) is a geodesic metric space, there exists a geodesic path joining x and w in X. We may assume that this path is parameterized by arc-length, let us denote it by γ : We have to prove that ≤ 0 for all nonnegative numbers a and b. If a = 0 and b = 0 this inequality is obvious. Let a and b be positive numbers. By Lemma 1, there exists some points x, y, z ∈ X such that d(x, y) = a, d(y, z) = b and where Ω is a proper simply-connected subdomain of C and ρ Ω is the hyperbolic metric on Ω, is a geodesic metric space and is unbounded. By Proposition 2 we get another proof for Theorem 1. Note that the pseudo-hyperbolic distance on Ω is not additive along geodesics of (Ω, ρ Ω ) [19].

The Case of Triangular Ratio Metric on a Canonical Plane Domain
The triangular ratio metric s G of a given proper subdomain G ⊂ R n is defined as follows for z 1 , Note that s G (z 1 , z 2 ) ≤ 1 for all z 1 , z 2 ∈ G. If no segment joining two points in G intersects the boundary ∂G, as it is the case if G is convex, then s G (z 1 , z 2 ) < 1 for all We have s G (z 1 , always attained. A point u ∈ ∂G is called a Ptolemy-Alhazen point for z 1 , z 2 ∈ G if a light ray from z 1 is reflected at u on the circle, such that the reflected ray goes through the point For a subset A of a convex domain G we may study the functional inequality where

The Triangular Ratio Metric on the Upper Half-Plane
Using the subadditivity of functions preserving the hyperbolic metric of the upper half-plane, we get the following

The Triangular Ratio Metric on the Unit Disk
There is an open conjecture stating that arctanhs D is a metric on the unit disk [23]. Note that arctanhs D is a metric on A ⊂ D if and only if We will prove that arctanh is metric-preserving with respect to the restrictions of the triangular ratio metric on the unit disk to radial segments, respectively, to circles centered at the origin.
Equality holds if and only if 0, x and y are collinear.

Lemma 3. The following addition formula holds
The restriction of arctanhs D to each diameter of the unit disk is a metric.
Proof. Recall that arctanh(u) = 1 2 log 1+u 1−u for all u ∈ (−1, 1). In particular, if −1 < u ≤ v < 1, then Since s D is invariant to rotations around the origin (more generally, s D is invariant to similarities), it suffices to prove that the restriction of arctanhs D to the intersection of the unit disk with the real axis is a metric. This follows from the above addition formula, observing that 0 ≤ max(arctanhs D (r, s), arctanhs D (s, t)) ≤ arctanhs D (r, t).
We prove that the restriction of the triangular ratio metric of the unit disk s D to each radial segment of the unit disk takes all values between 0 and 1.
Proof. Let λ ∈ (0, 1) and r ∈ 0, 1 2 . Assume that s ∈ (r, 1). Then (2) If f is amenable and f • tanh is subadditive and nondecreasing on [0, ∞), then the restriction of f • s D to every diameter of the unit disk D is a metric.
As s D is invariant to rotations around the origin, we may assume that the restriction Denote tanh a = λ and tanh b = µ. Fix r ∈ 0, 1 2 . By Lemma 4, there exists s ∈ r, 1 2 such that s D (r, s) = λ. Applying again Lemma 4, we get t ∈ s, 1 2 such that s D (s, t) = µ. The addition formula arctanhs D (r, t) = arctanhs D (r, s) + arctanhs D (s, t) shows that arctanhs D (r, t) = a + b.
Since the restriction of is amenable, subadditive and nondecreasing self-mapping of [0, ∞); therefore, F is metric-preserving. By Lemma 3, the restriction of arctanhs D to every diameter of the unit disk D is a metric. Then the restriction of f • s D = F • arctanhs D to every diameter of the unit disk D is a metric.
If f is amenable, subadditive and nondecreasing on [0, 1), then f • s D is a metric on the entire unit disk D and obviously f • tanh is amenable, subadditive and nondecreasing on [0, ∞).
The restriction of arctanhs D to each circle |z| = ρ < 1 is a metric.
Letting aside the case 2α = 2β = ω 0 , we get , attained at both endpoints of the interval.
The function G := F • tanh is metric-preserving. Therefore, F • s D = G • arctanhs D is a metric on the circle |z| = r.

Remark 6.
Triangle inequality for the restriction of arctanhs D to any circle |z| = r with r ∈ (0, 1) is always strict, as we see from the proof of Theorem 2. Therefore, given a, b ∈ (0, ∞) we cannot find x, y, z on a circle |z| = r, r ∈ (0, 1) such that arctanhs D (x, y) = a, arctanhs D (y, z) = b and arctanhs D (x, z) = a + b. This prevents us from obtaining the subadditivity of F • tanh under the assumption that F : [0, 1) → [0, ∞) with F −1 ({0}) = {0} is metric-preserving with respect to the restriction of the triangular ratio metric s D to every circle |z| = r < 1.

We prove a functional inequality similar to (4) satisfied by continuous functions
which are metric-preserving with respect to the restriction of the triangular ratio metric s D to every circle |z| = r < 1.
Computing sin(α + β) and cos(α + β), and using the notations ) and Let the function F : [0, 1) → [0, ∞) be amenable and metric-preserving with respect to the restriction of the triangular ratio metric s D to every circle |z| = r < 1.
Let r satisfying max(λ, µ) < r < 1. The triangle inequality F(d(x r , z r )) ≤ F(d(x r , y r )) + F(d(y r , z r )) may be written as We compute lim r 1 We have If F is continuous on [0, 1), then letting r tend to 1 from below in inequality (11) we get (9).

Remark 7.
Since sinh is supradditive and the function x → x √ 1+x 2 is increasing on R, we have sinh(a) + sinh(b) If F is nonincreasing, then inequality (10) implies the subadditivity of the function F • tanh on [0, ∞). If F is nondecreasing, then inequality (10) is implied by the subadditivity of the function F • tanh on [0, ∞).

The Case of Barrlund Metric with p = 2 on a Canonical Plane Domain
We will consider Barrlund metrics on canonical domains in plane: the upper half plane and the unit disk. For p = 2 and G ∈ {H, D} explicit formulas for b G,p have been proved in [23], as follows: Using parallelogram's rule, we can write We can see that b H,2 (H × H) = 0, √ 2 and b D,2 (D × D) = 0, √ 2 .
Next, we study the restrictions of b H,2 to vertical rays V x 0 :(Re(z) = x 0 and Im(z) > 0), x 0 ∈ R, to rays through origin O m : (Im(z) = mRe(z) and Im(z) > 0), m ∈ R * and to horizontal lines L c : (Im(z) = c), c ∈ (0, ∞). . The following are equivalent: (1) F is metric-preserving with respect to the restriction of b H,2 to every ray V x 0 , x 0 ∈ R; (2) F is metric-preserving with respect to the restriction of b H,2 to some ray V x 0 , x 0 ∈ R; (3) F • |ψ| is subadditive on R.
(2) ⇒ (3) Assume that F is metric-preserving with respect to the restriction of b H,2 to some fixed ray V x 0 .
(3) ⇒ (1) Assume that F • ψ is subadditive on R. Fix x 0 ∈ R. For every z k = x 0 + iy k ∈ V x 0 , k = 1, 2, 3 there exist u, v ∈ R such that y 2 y 1 = e u and y 3 y 2 = e v . Since u and v satisfy (13), we obtain the triangle inequality (12). It follows that the restriction of F • b H,2 to V x 0 is a metric, q.e.d.
. Then F is metric-preserving with respect to the restriction of b H,2 to the ray S m if and only if F • |κ m | is subadditive on R.
For z k = (1 + im)x k ∈ O m , k = 1, 2, 3 denote x 2 x 1 = e u and x 3 x 2 = e v , where u, v ∈ R. With these notations, the triangle inequality is equivalent to If F is metric-preserving with respect to the restriction of b H,2 to the ray O m , then for every u, v ∈ R we find z k = (1 + im)x k ∈ S m , k = 1, 2, 3 such that x 2 x 1 = e u and x 3 x 2 = e v and applying (14) we get (15). Conversely, if F • κ m is subadditive on R, then for every x 1 = e u and x 3 x 2 = e v and applying (14) we get (15).
We see that F • |ϕ c | satisfies (16) if and only if F • |ϕ c | is subadditive on R.
Next, we study metric-preserving functions with respect to the restriction of the Barrlund distance on the unit disk b D,2 to some one-dimensional manifolds, such as radial segments, diameters or circles centered at origin.
. The following are equivalent: (1) F is metric-preserving with respect to the restriction of b D,2 to every radial segment in the unit disk; (2) F is metric-preserving with respect to the restriction of b D,2 to some radial segment in the unit disk; (3) F • |ψ| is subadditive on R.
Proof. Obviously, (1) ⇒ (2). Using the invariance of the Barrlund distance on the unit disk with respect to rotations around the origin, it follows that (2) ⇒ (1), since (1) holds if and only if F is metric-preserving with respect to the restriction of b D,2 to the intersection I = [0, 1) between the unit disk and the non-negative semiaxis.
In order to prove that (2) and (3) are equivalent, we may assume without loss of generality that the radial segment in (2) is I = [0, 1). For z 1 = x 1 , z 2 = x 2 ∈ I, denoting For z k = x k ∈ I, k = 1, 2, 3 denote 1−x 2 1−x 1 = e u and 1−x 3 1−x 2 = e v , where u, v ∈ R. With these notations, the triangle inequality is equivalent to Assume that F is metric-preserving with respect to the restriction of b D,2 to the radial segment I. For every u, v ∈ R we find z k = x k ∈ I, k = 1, 2, 3 such that 1−x 2 1−x 1 = e u and 1−x 3 1−x 2 = e v . Indeed, we may choose any x 1 between max{0, 1 − e −u , 1 − e −u−v } and 1. Then it follows that 0 < x 3 < 1. Now applying (17) we get (18).
Conversely, if F • |ψ| is subadditive on R, then for every z k = x k ∈ I, k = 1, 2, 3 we find u, v ∈ R such that 1−x 2 1−x 1 = e u and 1−x 3 1−x 2 = e v and applying (18) we get (17). We give a sufficient condition for a function to be metric-preserving with respect to the restriction of the Barrlund metric b D,2 to some diameter of the unit disk, under the form of a functional inequality.
Assume that the restriction of F • b D,2 to some diameter of the unit disk is a metric. Then Proof. Since b D,2 is invariant to rotations around the origin, if a function is metric-preserving with respect to the restriction of the Barrlund metric b D,2 to some diameter of the unit disk, then that function is metric-preserving with respect to the restriction of the Barrlund metric b D,2 to every diameter of the unit disk. We may assume that the given diameter is on the real axis.
The above inequality writes as which is true, due to the assumption that the restriction of F • b D,2 to the diameter {x + i0 : −1 < x < 1} of the unit circle is a metric.

Conclusions
In this paper, we investigated properties related to subadditivity of the functions transferring some special metrics to metrics, establishing connections between metric geometry and functional inequalities. For a metric space, (X, d) such that d(x, y) ∈ [0, T) for all x, y ∈ X, where 0 < T ≤ ∞, let us denote by MP(X, d) the class of functions f : [0, T) → R + = [0, ∞) with the property that f • d is a metric on d. It is known from the theory of metric-preserving functions that the intersection of all classes MP(X, d) includes the class of all nondecreasing subadditive self-maps on R + and is included in the class of all subadditive self-maps on R + . We obtained functional inequalities satisfied by functions in MP(X, d) in several cases, where X is some subset of G ∈ {H, D} and d is the restriction to X of an intrinsic metric on G, namely the hyperbolic metric, the triangular ratio metric s G or the Barrlund metric b G,2 . We will denote by Sa([0, T)) the class of functions f : [0, T) → R + that are subadditive. In addition, denote by X r the circle of radius r ∈ (0, 1) centered at origin.
We summarize in Table 1 most of our results, excepting Proposition 2 and Theorem 3, namely Theorem 1 and Propositions 3, 4, 5, 6, 7, 8 and 10. Within the table we use the abbreviation F =MP(X, d): We determined the functions ψ (that is fixed), κ m , ϕ c and θ r (each depending only on the respective parameter).
Moreover, denoting d r = s D | X r , we proved that every f ∈ r∈(0,1) MP(X r , d r ) satisfies the functional inequality (10) related to the subadditivity of f • tanh, as follows. If f is nonincreasing on [0, 1) and satisfies (10), then f • tanh is subadditive on R + . If f is nondecreasing on [0, 1) and f • tanh is subadditive on R + , then f satisfies (10).