Inequalities for the generalized point pair function

We study a new generalized version of the point pair function defined with a constant α > 0. We prove that this function is a quasi-metric for all values of α > 0 and compare it to several hyperbolic-type metrics, such as the j∗-metric, the triangular ratio metric, and the hyperbolic metric. Most of the inequalities presented here have the best possible constants in terms of α. Furthermore, we research the distortion of the generalized point pair function under conformal and quasiregular mappings.


Introduction
Several different metrics can be used to study conformal [10], quasiconformal [5], quasiregular, or other mappings [11], which are an important subject of study in the geometric function theory.In the plane, the hyperbolic metric is useful for this purpose because of its invariance properties but, unfortunately, it can be defined only in special cases in dimensions n ≥ 3.For this reason, researchers have introduced numerous new hyperbolic-type metrics [3,9,14], which are designed after the hyperbolic metric so that they can measure the distances between points by taking their location with respect to the domain boundary into account.
Let G R n be a domain.For all points x ∈ G, denote the Euclidean distance to the boundary by d G (x) = inf z∈∂G |x − z|.For α > 0, define then the function p α G : G × G → [0, 1), [2, (5.1), p. 1391] This function above is called the generalized point pair function.It is derived from the point pair function p G , whose expression coincides with the special case α = 4 of the definition (1.1).The point pair function was originally introduced in [1] and studied further in [2,8,12,13,15].It was observed to be a useful tool for creating bounds for the hyperbolic metric and proved to be a quasi-metric for all domains G R n with the constant less than or equal to √ 5/2 [2, Thm 4.14, p. 1388].In 2022, Dautova et al. [2] introduced the generalized point pair function by replacing the constant 4 in the definition of the point pair function by a more general constant α > 0.
Considering the point pair function is very well-justified if the domain G is the upper half-space H n = {x = (x 1 , ..., x n ) ∈ R n | x n > 0}.This is because, for all points x, y ∈ H n , the distance p H n (x, y) in the point pair function is equal to the distance th(ρ H n (x, y)/2), where th is the hyperbolic tangent and ρ H n (x, y) is the hyperbolic metric defined in H n .However, defining this function with another constant instead of 4 might be more reasonable in some domains, which is why studying the generalized point pair function for values of α > 0 is useful.For instance, if 0 < α ≤ 12, it is known that the generalized point pair function is a metric in the domains R + [2, Thm 5.2, p 1391], R n \ {0} [2, Thm 5.11, p. 1395], and H n [2, Thm 5.13, p. 1396].Consequently, our aim is study this function further by comparing it to several hyperbolic-type metrics.
The structure of this article is as follows.First, in Section 2, we give the necessary notations and definitions.In Section 3, we study the inequality between the generalized point pair function and the hyperbolic-type metric known as the j * -metric, and prove that the generalized point pair function is a quasi-metric in every domain G R n .In Section 4, we give the similar inequalities for the triangular ratio metric and the t-metric.Finally, in Section 5, we present the inequalities between the generalized point pair function and the hyperbolic metric and use them to find some results for the distortion of the distances in the generalized point pair function under conformal and quasiregular mappings. of the triangle inequality with a constant c independent of the choice of the points x, y, z, then we call it a quasi-metric.Note that this type of a function is sometimes referred as a semi-metric, a metametric, or an inframetric instead.

Recall that a function
The Euclidean open ball with a center x ∈ R n and a radius r > 0 is denoted as B n (x, r) and its sphere is Define then the original point pair function [1, p. 685], [8, 2.4, p. 1124 The generalized point pair function is as in (1.1).To avoid possible confusion, note that p G means that α = 4 and, if α is unspecified, we mean the generalized version p α G .Next, consider the following hyperbolic-type metrics.The distance ratio metric introduced by Gehring and Osgood [4] is defined as This expression can be modified as in [ The triangular ratio metric, originally introduced by P. Hästö in 2002 [9], .
Furthermore, the t-metric defined as t G : G × G → [0, 1), was recently introduced in [14] but it must be noted that, unlike the distance ratio metric or the triangular ratio metric, this metric is not necessarily hyperbolic-type metric because the closures of its balls are not always compact in the domain G.
Use notations sh, ch and th for the hyperbolic sine, cosine, and tangent.Denote the upper half-space {x = (x 1 , ..., x n ) ∈ R n | x n > 0} by H n and use the notation B n for the Poincaré unit ball {x ∈ R n | |x| < 1}.In these two domains, the hyperbolic metric has the following formulas [6, (4.8) In the two-dimensional disk, we have where y is the complex conjugate of y.

Inequalities with the j * -metric
In this section, we first find the inequalities between the generalized point pair function and the j * -metric, then study the sharpness of the established inequalities, and use them to prove that the generalized point pair function is a quasi-metric.Theorem 3.1.For all x, y ∈ G R n and α > 0, the inequality holds.For the domain G = H n , these constants are the best ones possible in terms of α.
In fact, the first constant here is the best one possible in terms of α for every choice of the domain G R n .
Proof.By symmetry, we can fix distinct points x, y ∈ G such that d G (x) ≤ d G (y). Now, Let us first find the minimum of the quotient (3.2) in the case We see that the stationary point above is a minimum and, since this quotient is α/(α + 4) at |x − y| = αd G (x)/2, this is the minimum of the quotient (3.2).
The theorem follows from this, though note that we consider the reciprocals of the found extreme values since the bounds are presented for the function p α G (x, y).While the latter constant in Theorem 3.1 is not sharp for some choices of G R n , it follows from the next result that this constant is the best possible one in several common domains such as B n , H n , and R n \ ({0} ∪ {1}).Lemma 3.3.For G R n and α > 0, the constant (α + 4)/α is the best possible constant c in terms of α such that the inequality p α G (x, y) ≤ cj * G (x, y) holds for all points x, y ∈ G if (1) G contains an open ball so that the end points of one of its diameters belong to the boundary ∂G, or, (2) G contains an open half-ball but one of its diameters is fully on the boundary ∂G.
Proof.The inequality holds with c = (α + 4)/α according to Theorem 3.1 and it can be trivially verified that the equality holds here if Consider the first case where there are some points Suppose then that for q ∈ ∂G, r > 0, and h ∈ S n−1 (q, r), the half-ball Fix , so the result follows.
Lemma 3.4.For all x, y ∈ R n \ {0} and α > 0, the inequalities hold with the best possible constants in terms of α.
Proof.The left sides of both of the inequalities follow from Theorem 3.1, according to which they also have the best possible constants.By symmetry, assume that |x| ≤ |y| for the points x, y ∈ G = R n \ {0}.Let k be the angle between the vectors from the origin to x and to y.By writing the distance |x − y| with law of cosines, we will have To prove the right side of the inequalities in the lemma, we need to find the maximum value of this quotient.
By differentiation, This stationary point is a maximum.It fulfills −1 ≤ cos(k) ≤ 1 if and only if Consequently, the quotient (3.7) is increasing with respect to |y| and its maximum has a limit value 1 obtained when |y| → ∞.Thus, the result follows.
Corollary 3.8.For all G R n and α > 0, the function p α G (x, y) is a quasi-metric with a constant (α + 4)/α if α ≤ 4 and 2 Proof.It follows from Theorem 3.1 and the fact that j * G (x, y) is a metric that Note that, for the domain G = R n \ {0}, the inequalities of Lemma 3.4 would give better constants for Corollary 3.8, but it is already proven that the generalized point pair function is a metric in R n \ {0} for all 0 < α ≤ 12 [2, Thm 5.11, p. 1395].

Inequalities with the triangular ratio metric and the t-metric
Let us now find the inequalities between the generalized point pair function and the triangular ratio metric and the t-metric.Lemma 4.1.For all x, y ∈ G R n and β > α > 0, the inequality holds with the best possible constants in terms of α.
Proof.Clearly, the quotient attains its minimum value 1 when either x or y approaches boundary so that d G (x) → 0 + or d G (y) → 0 + , and its maximum value β/α when the points x and y approach to each other so that |x − y| → 0 + .Lemma 4.3.For all x, y ∈ G R n and α > 0, and, if G is convex, the left sides of these inequalities can be improved by replacing constants Let us now combine the inequalities (4.4) and (4.6).Note that The first part of the lemma follows from this.Suppose then that G is convex.By [8, Thm 2.9(i), p. 1129], s G (x, y) ≤ √ 2j * G (x, y) holds in this case so it follows from Theorem 3.1 that Furthermore, s G (x, y) ≤ p G (x, y) in a convex domain G by [6, lemma 11.6(1), p. 197], so it follows from the inequality (4.5) that min 1, √ α The rest of the lemma follows from the two inequalities above as Lemma 4.7.For all x, y ∈ G R n and α > 0, the inequalities hold with the best possible constant in terms of α.
Proof.Consider the quotient The stationary point above is a minimum.By symmetry, let us assume that so the expression is increasing with respect to |x − y| when |x − y| ≤ (αd G (x))/2.It has a limit value Consider then the case |x − y| > αd G (x)/2.Now, the quotient (4.8) is increasing with Consequently, the quotient (4.8) is monotonic with respect to |x − y|.The quotient (4.8) has a limit value 1 + 4/α when |x − y| → αd G (x)/2 and a limit value 1 when |x − y| → ∞, out of which 1 is smaller.Thus, it follows that the infimum of the quotient (4.8) is min{1, 2/ √ α}.It follows from the earlier differentiation of the quotient (4.8) that it is at maximum with respect to d G (y) in one of the end points of the interval , the quotient (4.8) is the quotient (4.9), which was noted to be monotonic with respect to |x − y|.The maximum value of the quotient (4.9) has either a limit value 2/ √ α obtained when |x − y| → 0 The derivative above is positive if either α ≥ 2 or α < 2 and |x − y| < αd G (x)/(2 − α).Consequently, if α ≥ 2, the quotient (4.10) is increasing with respect to |x − y| and its maximum has a limit value 2 obtained when |x − y| → ∞.If α < 2 instead, the maximum of the quotient (4.10) is 4/ α(4 − α) at |x − y| = αd G (x)/(2 − α).Because these values are greater than the limit values of the maximum values of the quotient (4.9), it follows that the supremum of the quotient (4.8) is either 4/ α(4 − α) if α < 2 and 2 if α ≥ 2. The inequalities of the lemma now follow.
The limit value 2/ √ α of the quotient (4.8) can be obtained in any domain G R n by choosing y ∈ B n (x, d G (x)) for any fixed point x ∈ G so that d G (y) → d G (x) and |x − y| → 0 + .Similarly, the limit value 1 can be found by choosing x, y ∈ G so that d G (x), d G (y) → 0 + .If α < 2, the value 4/ α(4 − α) of the quotient (4.8) is possible to find by fixing x ∈ G, z ∈ S n−1 (x, d G (x)) ∩(∂G), and y = x+ α(z −x)/2.Furthermore, the limit value 2 of the quotient (4.8) can be obtained if we fix It follows from this that we have the best constants in terms of α, regardless of the choice of the domain G.

Inequalities with the hyperbolic metric
In this section, we first study the inequalities between the generalized point pair function and the hyperbolic metric in the upper half-space and the unit ball, and then study the distortion of the generalized point pair function under Möbius and quasiregular mappings.
Corollary 5.1.For all x, y ∈ H n and α > 0, the inequality holds with the best possible constants in terms of α.
Proof.The results follows from Lemma 4.1 and the fact that th(ρ H n (x, y)/2) = p H n (x, y) by [6, p. 460].
Theorem 5.2.For all x, y ∈ B n and α > 0, the inequality holds with the best possible constants in terms of α.
Proof.The values of p α B n (x, y) and ρ B n (x, y) only depend on how the points x, y are fixed on the intersection of the unit ball and the two-dimensional plane containing x, y, and the origin, so it is enough to prove this inequality in case n = 2 by studying the quotient It follows that, if α < 2, the maximum of the quotient (5.4) is 2/ α(4 − α).Otherwise, the maximum is 1.The minimum of the quotient (5.4) is also 1 or 1/ √ α, depending if α < 1 or not.By symmetry, these are the extreme values of the quotient (5.3) also in the case x = 0.
Suppose then that x = 0 = y.Let k ∈ [0, π] be the angle between the vectors from the origin to x and y, or equivalently k = Arg(x/y).By law of cosines, Consequently, the quotient (5.3) can be written as Thus, the quotient (5.5) is monotonic with respect to cos(k) and is at minimum when cos(k) = −1 and at maximum when cos(k) = 1 or vice versa, depending on if α < (1 + |x|)(1 + |y|) or not.
Let us first consider the case cos(k) = −1.Now, the quotient (5.5) becomes We see that the only stationary point of the quotient (5.6) with respect to |y| is a maximum.However, the maximum of this quotient (5.6) is the maximum of the quotient (5.5) if and only if α ≥ (1 + |x|)(1 + |y|).Because the stationary point fulfills it cannot be the maximum of the quotient (5.5).Thus, the quotient (5.6) can offer extreme values of the quotient (5.5)only when |y| → 0 + or |y| → 1 − .The case y = 0 was already considered earlier and, if |y| → 1 − , the quotient (5.6) approaches 1.
Let us next consider the case cos(k) = 1, where the quotient (5.5) is The quotient (5.9) is 1 at and, since the inverse mapping f −1 of any conformal mapping is another conformal mapping, the first part of inequality follows directly from this.
Consider the Möbius transformation T a : B 2 → B 2 , defined as T a (z) = (z − a)/(1 − az).It has been observed that the Lipschitz constant of this mapping seems to be 1 + |a| for several intrinsic metrics and quasi-metrics defined in the unit disk, including the triangular ratio metric [1, Conj.1.6, p. 684], the j * -metric [12], the t-metric [14,Conj. 4.4], the point pair function [12], and the Barrlund metric [3,Conj. 4.3,p. 25].Computer tests suggest that this also holds for the generalized point pair function, regardless of the value of α > 0.
1) d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y, (2) d(x, y) = d(y, x), and (3) d(x, y) ≤ d(x, z)+d(z, y).The third one of these properties is called the triangle inequality.If a function fulfills the two first properties and the relaxed version d(x, y) ≤ c(d(x, z) + d(z, y))

. 2 )
Clearly, for fixed choices of d G (x) and |x − y|, this quotient is increasing with respect to d G (y).Because of the triangle inequality, d G (y) ≤ d G (x) + |x − y|, so the value of d G (y) is limited to the closed interval from d G (x) to d G (x) + |x − y|.Consequently, the quotient (3.2) is at minimum with respect to d G (y) when d G (y) = d G (x) and at maximum when d G (y) = d G (x) + |x − y|.