Stabilizing Metrics Induced by Logarithmic Oscillation in Metric Spaces and Gromov Hyperbolicity

1. Introduction

Gromov hyperbolicity, introduced in [12], is a central concept in metric geometry, providing a broad generalization of negatively curved spaces . Gromov hyperbolic spaces exhibit rich geometric and topological features, including thin triangles in geodesic metric spaces and a well-defined boundary at infinity. Hyperbolic-type metrics have permeated several areas of mathematics, such as geometric function theory [13], metric geometry and geometric group theory [7], convex geometry [29], dynamical systems [8], and have been recently applied in theoretical computer science via graph theory and network analysis [9]. Hyperbolic geometry offers a powerful alternative to Euclidean geometry for representing complex networks and graph data arising in data science and machine learning, allowing large-scale, hierarchical data to be embedded with significantly lower distortion and reduced dimension. For a mini-monograph on Gromov hyperbolic metric spaces which need not be geodesic, from the point of view of geometric function theory, see [31]. In 2007 Lindén [21] gave a status report on the research of various hyperbolic-type metrics in a proper subdomain G of ${\mathbb{R}}^n$. Since then, many new hyperbolic-type metrics have been introduced and studied, such as the triangular ratio metric [15], the Cassinian metric [18], the visual angle metric [19], the metric of Dovgoshey, Hariri and Vuorinen [11], the Nikolov-Andreev metric $i_G$ [28], the Ibragimov metric [17], two types of stabilizing metrics [6], a modified distance ratio metric [24] etc. The Gromov hyperbolicity of such metrics has been studied in [23,35,36] and many other recent research papers. In [25,26] some generalizations of the Gehring-Osgood metric, Vuorinen’s distance ratio metric, the metric of Dovgoshey, Hariri and Vuorinen, the Nikolov-Andreev metric and the Ibragimov metric have been investigated in the setting of general metric spaces, including from the perspective of Gromov hyperbolicity.

Many hyperbolic-type metrics in subdomains of of ${\mathbb{R}}^n$and their applications to geometric function theory have been thoroughly studied in the monographs of Anderson, Vanamurthy and Vuorinen [1] and of Hariri, Klén and Vuorinen [13]. See also [10] to understand the place of hyperbolic-type metrics in the context of general theory of metric spaces.

Barbilian’s metrization procedure, which was introduced in [2] in order to provide a generalization of the Klein-Beltrami distance in hyperbolic geometry, was extended in an abstract setting by Boskoff, Ciuc ă and Suceavă [5], as follows. Given two arbitrary nonempty sets K and J, a function $f:K\times J\to (0,\infty )$ is called an influence of K over J if for every $x,y\in J$ the ratio $g_{x,y}\left(z\right)={\displaystyle \frac{f(z,x)}{f(z,y)}}$ with $z\in K$ has a maximum over K. If T is a topological space, $K\subset T$ is a compact subset and $f:K\times J\to (0,\infty )$ is continuous in the first argument, then f is an influence of K over the arbitrary nonempty set J [5]. Barbilian pointed out that the existence of $\underset{z\in K}{max}{g}_{x,y}\left(z\right)\in \mathbb{R}$ implies the existence of $\underset{z\in K}{min}{g}_{x,y}\left(z\right)\in \mathbb{R}$ and $\underset{z\in K}{min}{g}_{x,y}\left(z\right)={\displaystyle \frac{1}{\underset{z\in K}{max}{g}_{y,x}\left(z\right)}}$. More general, $\underset{z\in K}{sup}{g}_{x,y}\left(z\right)<\infty$ for $\left(x,y\right)\in J\times J$ implies $\underset{z\in K}{inf}{g}_{x,y}\left(z\right)={\displaystyle \frac{1}{\underset{z\in K}{sup}{g}_{y,x}\left(z\right)}}$. Assuming that $\underset{z\in K}{sup}{g}_{x,y}\left(z\right)<\infty$ for any $\left(x,y\right)\in J\times J$, it was shown in [5] that the function $d:J\times J\to [0,\infty )$ defined by $d(x,y)=log(\underset{z\in K}{sup}{g}_{x,y}\left(z\right)/\underset{z\in K}{inf}{g}_{x,y}\left(z\right))$ is a semi-distance, which extends Barbilian’s logarithmic oscillation. In particular, if for each pair $\left(x,y\right)\in J\times J$ the ratio $g_{x,y}$ is not constant on K, then d is a distance on J.

Barbilian’s metrization procedure yields various Riemannian and generalized Lagrangian metrics, see [5]. Also, the Apollonian distance on a bounded domain $G\subset {\mathbb{R}}^n$ with $card({\mathbb{R}}^n\setminus G)\ge 2$, studied by Beardon [3], is a result of Barbilian’s metrization procedure. The Apollonian semi-distance, defined by ${\alpha }_G(x,y)=\underset{a,b\in \partial G}{sup}log\left({\displaystyle \frac{\left|y-a\right|}{\left|x-a\right|}}{\displaystyle \frac{\left|x-b\right|}{\left|y-b\right|}}\right)$, is the Barbilian’s logarithmic oscillation determined by the influence function $f:\partial G\times G\to (0,\infty )$, $f(z,x)=\left|z-x\right|$, and is a distance provided that the boundary of G is not a subset of a sphere or of a hyperplane [21]. Note that the Apollonian metric coincides with the classical hyperbolic metric if $G={\mathbb{H}}^n$ [21] or $G={\mathbb{B}}^n$ [6,20][Theorem 3.5]. The Apollonian metric on a bounded domain in ${\mathbb{R}}^n$, satisfying the above assumptions, is Gromov hyperbolic, as it is roughly isometric with Gehring-Osgood distance ratio metric [21].

In the special case where $T={\mathbb{R}}^n$, $K=\left\{0_{{\mathbb{R}}^n}\right\}$, $J={\mathbb{R}}^n\setminus \left\{0\right\}$ and $f:K\times J\to (0,\infty )$, $f(z,x)=\left|z-x\right|$, the Barbilian’s logarithmic oscillation is given by $d_B(x,y)=log{\displaystyle \frac{max\left\{\left|x\right|,\left|y\right|\right\}}{min\left\{\left|x\right|,\left|y\right|\right\}}}$ and is a Gromov hyperbolic semi-distance with $\delta =0$ [6], Theorem 4.1. In [6], Boskoff and Suceavă introduced two new metrics induced by the logarithmic oscillation $d_B$, called stabilizing metrics, given by $d_1(x,y)=log\left({\displaystyle \frac{max\left\{\left|x\right|,\left|y\right|\right\}}{min\left\{\left|x\right|,\left|y\right|\right\}}}+\left|x-y\right|\right)$ and $d_2(x,y)=log\left({\displaystyle \frac{max\left\{\left|x\right|,\left|y\right|\right\}}{min\left\{\left|x\right|,\left|y\right|\right\}}}+{\displaystyle \frac{\left|x-y\right|}{min\left\{\left|x\right|,\left|y\right|\right\}}}\right)$, for $x,y\in {\mathbb{R}}^n\setminus \left\{0\right\}$. The term added to the oscillation ${\displaystyle \frac{max\left\{\left|x\right|,\left|y\right|\right\}}{min\left\{\left|x\right|,\left|y\right|\right\}}}$ ensures that $d_1$ and $d_2$ are genuine metrics on any set $G\subset {\mathbb{R}}^n\setminus \left\{0\right\}$, which are less sensitive to the influence of the origin than Barbilian’s logarithmic oscillation $d_B$. Boskoff and Suceavă proved that the metrics $d_1$ and $d_2$ are Gromov hyperbolic with a constant $\delta \le log3$ on the punctured unit ball $G={\mathbb{B}}^n\setminus \left\{0\right\}$. They also introduced and studied generalizations of the stabilizing metrics $d_1$, $d_2$ and of Vuorinen’s distance ratio metric $j_G$, as shown below. Let $M{\mathbb{R}}^n$ and $G\subset {\mathbb{R}}^n\setminus M$ be nonempty sets and let $f:[0,\infty )\to [0,\infty )$ be a non-decreasing subadditive function, such that $f\left(x\right)=0$ if and only if $x=0$, and $\beta x\le f\left(x\right)\le \alpha x$ for all $x\in [0,\infty )$, for some constants $\alpha \ge \beta >0$. Denote ${\delta }_M\left(x\right)=dist(x,M)=inf\left\{\left|z-x\right|:z\in M\right\}$. The generalizations of $d_1$, $d_2$ and $j_G$ [6] are defined, for $x,y\in {\mathbb{R}}^n\setminus \overline{M}$, by $d_f(x,y)=log\left({\displaystyle \frac{max\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}{min\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}}+f\left(\left|x-y\right|\right)\right)$, $d_{VH,s}=log\left({\displaystyle \frac{max\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}{min\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}}+{\displaystyle \frac{\left|x-y\right|}{min\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}}\right)$ and $d_{V,f}(x,y)=log\left(1+{\displaystyle \frac{f\left(\left|x-y\right|\right)}{min\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}}\right)$. Note that ${\delta }_M\left(z\right)=0$ is equivalent to $z\in \overline{M}$, therefore the above expressions are well-defined for all $x,y\in {\mathbb{R}}^n\setminus M$ if and only if the supporting set M is closed.

The purpose of this paper is to study some extensions of the stabilizing metrics introduced by Boskoff and Suceavă [6], obtained by passing from the setting of the Euclidean space ${\mathbb{R}}^n$ to that of an arbitrary metric space $(X,d)$ and by replacing the distance ${\delta }_M$ to $M\subset X$ by an abstract influence function $F:X\setminus M\to (0,\infty )$, as in [25] and [26]. This allows for more flexibile choices in possible applications of stabilizing metrics. Throughout this paper, the modifying function $f:[0,\infty )\to [0,\infty )$ is non-decreasing, subadditive and vanishes only at the origin. In Section 3 we study a generalization of Barbilian’s logarithmic oscillation $d_B$, defined by $d_{B,F}\left(x,y\right)=log{\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}$ and an associated stabilizing distance function defined by

$$d_{M,F,f}\left(x,y\right)=log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}+f\left(d(x,y)\right)\right),$$

(1)

for $x,y\in {\mathbb{R}}^n\setminus M$. We prove that $d_{B,F}$ is a semi-distance on $X\setminus M$, with Gromov hyperbolicity constant $\delta =0$, while $d_{M,F,f}$ is a metric on $X\setminus M$. In Section 4 the Gromov hyperbolicity of the the stabilizing metric $d_{M,F,f}$ is investigated, in the special case where $M=\left\{p\right\}$ is a singleton and $F\left(x\right)=d(x,p)$. If $(X,d)$ is Ptolemaic and f is the identity of $(0,\infty )$, the stabilizing metric $d_{M,F,f}$ is Gromov hyperbolic with a constant $\delta \le log2$. This result is a strong generalization of [6], Theorem 4.2 (ii), with an improvement of the Gromov hyperbolicity constant; it would be interesting to investigate if $d_{M,F,f}$ is strongly hyperbolic in this case. If there exists $\lambda >0$ such that $f\left(t\right)\le \lambda t$ for all $t\in [0,\infty )$ and $G\subset X\setminus \left\{p\right\}$ is a bounded set, then the restriction of $d_{M,F,f}$ to $G\times G$ is Gromov hyperbolic with $\delta \le log\left(1+2\lambda K\right)$, where $K=\underset{z\in G}{sup}d(z,p)\le \mathrm{dist}(p,G)+\mathrm{diam}\left(G\right)$; this result extends [6], Theorem 5.1. (b). We also study generalizations of Vuorinen’s distance ratio metric and of Vuorinen’s stabilizing metric $d_{VH,s}$, defined by $j_{F,f}\left(x,y\right)=log\left(1+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\right)$, respectively by $j_{F,f,s}\left(x,y\right)=log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\right)$, $x,y\in X\setminus M$. If F is $1-$Lipschitz with respect to the metric $f\circ d$, we prove that $j_{F,f}$ and $j_{F,f,s}$ are metrics on $X\setminus M$, with $j_{F,f}\le j_{F,f,s}\le 2j_{F,f}$. We provide sufficient conditions for the Gromov hyperbolicity of $j_{F,f}$, generalizing [14], Theorem 3, Sufficiency and [6], Theorem 6.3 (b), respectively of $j_{F,f,s}$, generalizing [6], Theorem 6.1 (b).

For each metric $\rho \in \left\{d_{M,F,f},j_{F,f},j_{F,f,s}\right\}$, under some natural assumptions on F and f, we show that the metric space $(X\setminus M,\rho )$ is complete provided that $(X,d)$ is complete and we give an upper bound for the linear dilatation of the identity map $1_{X\setminus M}:(X\setminus M,d)\to (X\setminus M,\rho )$.

2. Preliminaries

We recall that an intrinsic metric space (or length space) is a metric space where the distance between any two points equals the infimum of the lengths of all rectifiable curves connecting them. A geodesic metric space is an intrinsic metric space in which the above infimum is achieved for every pair of points.

In a general metric space $\left(X,\rho \right)$ Gromov hyperbolicity is defined using the Gromov product, as follows. The Gromov productof $x,y\in X$ with respect to a base point $w\in X$ is ${\left(\left.x\right|y\right)}_w={\displaystyle \frac{1}{2}}\left(\rho (x,w)+\rho (y,w)-\rho \left(x,y\right)\right)$. The metric space $\left(X,\rho \right)$ is said to be Gromov hyperbolic with base point $w\in X$ if there exists a constant $\delta \ge 0$ such that

$${\left(\left.x\right|z\right)}_w\ge min\left\{{\left(\left.x\right|y\right)}_w,{\left(\left.y\right|z\right)}_w\right\}-\delta$$

(2)

for every $x,y,z\in X$ [14]. If the inequality (2) holds for some $w\in X$ with a constant $\delta$, then it holds for every $w\in X$ with $\delta$ replaced by $2\delta$. Equivalently, the metric space is Gromov hyperbolic if there exists a constant $\delta \ge 0$ such that, for all $x,y,z,w\in X$,

$$\rho \left(x,z\right)+\rho \left(y,w\right)\le max\left\{\rho \left(x,w\right)+\rho \left(y,z\right),\rho \left(x,y\right)+\rho \left(z,w\right)\right\}+2\delta .$$

(3)

If the above inequality holds for all $x,y,z,w\in X$, one says that the metric space $(X,\rho )$ is Gromov hyerbolic with a constant $\delta$, shortly, $(X,\rho )$ is $\delta -$hyerbolic. Inequality (3) makes sense also for a semi-distance (pseudo-metric) $\rho$. A semi-distance$\rho :X\times X\to \mathbb{R}$ is a function satisfying the axioms of non-negativity, symmetry, the triangle inequality, and the condition $\rho (x,x)=0$ for all $x\in X$, but not necessarily the condition that $\rho (x,y)=0$ implies $x=y$.

An enhancement of the notion of Gromov hyperbolic space is that of strongly hyperbolic space, introduced in 2016 by Nica and Špakula [27]. A metric space $(X,\rho )$ is said to be strongly hyperbolic with parameter $\epsilon >0$ if

$$e^{{\displaystyle \frac{\epsilon }{2}}(\rho \left(x,z\right)+\rho \left(y,w\right))}\le e^{{\displaystyle \frac{\epsilon }{2}}(\rho \left(x,w\right)+\rho \left(y,z\right))}+e^{{\displaystyle \frac{\epsilon }{2}}(\rho \left(x,y\right)+\rho \left(z,w\right))}$$

for all $x,y,z,w\in X$. One can see that every strongly hyperbolic metric space with parameter $\epsilon >0$ is Gromov hyperbolic with a constant $\delta \le {\displaystyle \frac{log2}{\epsilon }}$ [27], Tjeorem 4.2. It is known that $CAT(-1)$ spaces, in particular the hyperbolic half-plane ${\mathbb{H}}^2$, are strongly hyperbolic with parameter $\epsilon =1$.

The concept of Ptolemaic metric space plays an important role in the theory of hyperbolic-type metrics. A metric space $(X,\rho )$ is said to be Ptolemaic if it satisfies the following Ptolemy inequality

$$\rho (x_1,x_3)\rho \left(x_2,x_4\right)\le \rho (x_1,x_2)\rho \left(x_3,x_4\right)+\rho (x_1,x_4)\rho \left(x_2,x_3\right).$$

for all $x_1,x_2,x_3,x_4\in X.$

A normed space is Ptolemaic if and only if its norm is induced by an inner product [30]. Also, all $CAT\left(0\right)$ spaces are Ptolemaic (see [34] and the references therein).

An interesting connection between Ptolemaic metric spaces and strongly hyperbolic metric spaces has been established by Zhang and Xiao [34], Theorem 2: if the metric space $(X,d)$ is Ptolemaic, then $(X,log(1+d\left)\right)$ is a strongly hyperbolic metric space with parameter $\epsilon =2$.

In order to study the changes of the quasiconformal geometry induced by the change of the original metric with a hyperbolic-type metric we need the following metric definition of quasiconformal maps between metric spaces [16]. Given a homeomorphism f from a metric space $\left(X,d\right)$ to a metric space $\left(Y,d^{\prime }\right)$, then for $x\in X$ and $r>0$, the linear dilatation off at x is defined by

$$H_f\left(x,r\right)={\displaystyle \frac{sup\left\{d^{\prime }(f\left(x\right),f\left(y\right)):d\left(x,y\right)\le r\right\}}{inf\left\{d^{\prime }(f\left(x\right),f\left(y\right)):d\left(x,y\right)\ge r\right\}}}.$$

A homeomorphism $f:\left(X,d\right)\to \left(Y,d^{\prime }\right)$ is called $H-$quasiconformal, with a nonnegative constant $H<\infty$, if $\underset{r\to 0}{limsup}{H}_f\left(x,r\right)\le H$ for every $x\in X$.

Some properties of a function $f:[0,\infty )\to [0,\infty )$ that vanishes only at the origin, is nondecreasing and subadditive will be useful in the sequel.

Lemma 1.  

Let $f:[0,\infty )\to [0,\infty )$ be a function satisfying $f\left(0\right)=0$.

(a) If f vanishes only at the origin and is nondecreasing, then $\underset{n\to \infty }{lim}f\left(t_n\right)=0$ implies $\underset{n\to \infty }{lim}{t}_n=0.$

(b) If f is subadditive and differentiable at the origin, then $f\left(t\right)\le f^{\prime }\left(0\right)t$ for all $t\in [0,\infty )$.

(c) If f is nondecreasing, subadditive and f is continuous at the origin, then f is continuous on $[0,\infty )$.

Proof. (a) Assume by contrary that there is a sequence ${\left(t_n\right)}_{n\ge 1}$ in $[0,\infty )$ that does not converge to the origin, such that $\underset{n\to \infty }{lim}f\left(t_n\right)=0$. Then there exists a subsequence ${\left(t_{n_k}\right)}_{k\ge 1}$ such that $t:=\underset{k\ge 1}{inf}{t}_{n_k}>0$. As f is nonnegative, nondecreasing and vanishes only at the origin, for all positive integers k we have $f\left(t_{n_k}\right)\ge f\left(t\right)>0$, a contradiction with $\underset{n\to \infty }{lim}f\left(t_n\right)=0$.

(b) For $t=0$ the claim holds, since $f\left(0\right)=0$.

Using the subadditivity of f, for every $x\in [0,\infty )$ and all positive integers n, we get $f\left(x\right)\le nf\left({\displaystyle \frac{x}{n}}\right)$. For $x>0$ this implies

$${\displaystyle \frac{f\left(x\right)}{x}}\le {\displaystyle \frac{f(x/n)}{x/n}}.$$

Taking into account that $f\left(0\right)=0$ and there exists $f^{\prime }\left(0\right)\in \mathbb{R}$, letting n tend to infinity in the above inequality, we obtain ${\displaystyle \frac{f\left(x\right)}{x}}\le f^{\prime }\left(0\right)$.

(c) Using the assumption that f is nondecreasing and subadditive, we obtain the inequality $\left|f\left(s\right)-f\left(t\right)\right|\le f\left(\left|s-t\right|\right)$ for all $s,t\in [0,\infty )$. For every convergent sequence ${\left(s_n\right)}_{n\ge 1}$ in $[0,\infty )$ with $\underset{n\to \infty }{lim}{s}_n=s$, as $\left|f\left(s_n\right)-f\left(s\right)\right|\le f\left(\left|s_n-s\right|\right)$ for all $n\ge 1$ and f has the limit zero at the origin, it follows that $\underset{n\to \infty }{lim}f\left(s_n\right)=f\left(s\right).$ □

Remark 1.  

The following sufficient condition for subadditivity on $\left(0,\infty \right)$ is well-known. If ${\displaystyle \frac{f\left(t\right)}{t}}$ is nonincreasing on $\left(0,\infty \right)$, then f is subadditive. Indeed, for all $s,t\in \left(0,\infty \right)$, ${\displaystyle \frac{f\left(s\right)}{s}}\ge {\displaystyle \frac{f(s+t)}{s+t}}$ and ${\displaystyle \frac{f\left(t\right)}{t}}\ge {\displaystyle \frac{f(s+t)}{s+t}}$, hence $f\left(s\right)+f\left(t\right)\ge s{\displaystyle \frac{f(s+t)}{s+t}}+t{\displaystyle \frac{f(s+t)}{s+t}}=f(s+t).$

Lemma 2.  

If d is a metric on X and $f:[0,\infty )\to [0,\infty )$ is nondecreasing and subadditive, then $f\circ d$ is also a metric on X.

The most useful examples of nondecreasing subadditive functions $f:[0,\infty )\to [0,\infty )$ are $f\left(t\right)=t^{\alpha }$ with $\alpha \in (0,1]$ and $f\left(t\right)=log(1+t)$, some other examples being $f\left(t\right)=arcsin\mathrm{h}\left(t\right)$ and $f\left(t\right)=arccos\mathrm{h}(1+t)$ [15]. For every metric space $\left(X,d\right)$ and all $\alpha >0$, the metric space $\left(X,d^{\alpha }\right)$ is called a snowflake version of the given metric space.

3. Modified Barbilian’s Logarithmic Oscillation and Stabilizing Distance in an Arbitrary Metric Space

The following elementary Lemmas are folklore, so we provide short proofs for completeness.

Lemma 3.  

For all positive numbers $\alpha ,\beta ,\gamma$ the following inequality holds:

$${\displaystyle \frac{max\left\{\alpha ,\beta \right\}}{min\left\{\alpha ,\beta \right\}}}\le {\displaystyle \frac{max\left\{\alpha ,\gamma \right\}}{min\left\{\alpha ,\gamma \right\}}}{\displaystyle \frac{max\left\{\gamma ,\beta \right\}}{min\left\{\gamma ,\beta \right\}}}.$$

(4)

Equivalently, we have

$$max\left\{{\displaystyle \frac{\alpha }{\beta }},{\displaystyle \frac{\beta }{\alpha }}\right\}\le max\left\{{\displaystyle \frac{\alpha }{\gamma }},{\displaystyle \frac{\gamma }{\alpha }}\right\}max\left\{{\displaystyle \frac{\gamma }{\beta }},{\displaystyle \frac{\beta }{\gamma }}\right\}.$$

The equality holds if and only if $min\left\{\alpha ,\beta \right\}\le \gamma \le max\left\{\alpha ,\beta \right\}$.

Proof. 

We may assume without loss of generality that $\alpha \le \beta$. The inequality (4) has one of the following forms

Case 1. If $\gamma =min\left\{\alpha ,\beta ,\gamma \right\}$, ${\displaystyle \frac{\beta }{\alpha }}\le {\displaystyle \frac{\alpha }{\gamma }}{\displaystyle \frac{\beta }{\gamma }}$, i.e. ${\displaystyle \frac{\beta \left(\gamma +\alpha \right)\left(\gamma -\alpha \right)}{\alpha {\gamma }^2}}\le 0$.

Case 2. If $\gamma =max\left\{\alpha ,\beta ,\gamma \right\}$, ${\displaystyle \frac{\beta }{\alpha }}\le {\displaystyle \frac{\gamma }{\alpha }}{\displaystyle \frac{\gamma }{\beta }}$, i.e. ${\displaystyle \frac{\left(\beta +\gamma \right)\left(\beta -\gamma \right)}{\alpha \beta }}\le 0$.

Case 3.If $\alpha \le \gamma \le \beta$, ${\displaystyle \frac{\beta }{\alpha }}\le {\displaystyle \frac{\gamma }{\alpha }}{\displaystyle \frac{\beta }{\gamma }}$.

In all these three cases, the inequality (4) is checked and in the third case it holds as an equality. □

In the proofs of [6], Theorem 4.1. (2) and [14], Theorem 3, the following inequality has been used.

Lemma 4.  

For all positive numbers $\alpha ,\beta ,\gamma ,\omega$

$$max\left\{{\displaystyle \frac{\alpha }{\beta }},{\displaystyle \frac{\beta }{\alpha }}\right\}max\left\{{\displaystyle \frac{\gamma }{\omega }},{\displaystyle \frac{\omega }{\gamma }}\right\}\le max\left\{max\left\{{\displaystyle \frac{\alpha }{\gamma }},{\displaystyle \frac{\gamma }{\alpha }}\right\}max\left\{{\displaystyle \frac{\beta }{\omega }},{\displaystyle \frac{\omega }{\beta }}\right\},max\left\{{\displaystyle \frac{\alpha }{\omega }},{\displaystyle \frac{\omega }{\alpha }}\right\}max\left\{{\displaystyle \frac{\beta }{\gamma }},{\displaystyle \frac{\gamma }{\beta }}\right\}\right\}.$$

The inequality is sharp.

Proof. 

Without loss of generality, we may assume that $\alpha \le min\left\{\beta ,\gamma ,\omega \right\}$. In this case, the above inequality reduces to

$${\displaystyle \frac{\beta }{\alpha }}max\left\{{\displaystyle \frac{\gamma }{\omega }},{\displaystyle \frac{\omega }{\gamma }}\right\}\le max\left\{{\displaystyle \frac{\gamma }{\alpha }}max\left\{{\displaystyle \frac{\beta }{\omega }},{\displaystyle \frac{\omega }{\beta }}\right\},{\displaystyle \frac{\omega }{\alpha }}max\left\{{\displaystyle \frac{\beta }{\gamma }},{\displaystyle \frac{\gamma }{\beta }}\right\}\right\},$$

which is equivalent to

$$max\left\{{\displaystyle \frac{\beta \gamma }{\omega }},{\displaystyle \frac{\beta \omega }{\gamma }}\right\}\le max\left\{{\displaystyle \frac{\beta \gamma }{\omega }},{\displaystyle \frac{\gamma \omega }{\beta }},{\displaystyle \frac{\beta \omega }{\gamma }}\right\},$$

which obviously holds. The equality holds in this case if and only if $\alpha \le min\left\{\gamma ,\omega \right\}\le \beta$. □

Lemma 5.  

A metric space $(X,d)$ is Gromov hyperbolic with $\delta =0$ if and only if for every $\epsilon >0$ the space $(X,d)$ is strongly hyperbolic with parameter ε. Moreover, if there exist arbitrarly large values of ε such that $(X,d)$ is strongly hyperbolic with parameter ε, then $(X,d)$ is Gromov hyperbolic with $\delta =0$.

Proof.$(X,d)$ is Gromov hyperbolic with $\delta =0$ if and only if for all $x$, y, z, $w\in X$,

$$d\left(x,y\right)+d\left(z,w\right)\le max\left\{d\left(x,z\right)+d\left(y,w\right),d\left(x,w\right)+d\left(y,z\right)\right\}$$

(5)

For every $\epsilon >0$, inequality (5) implies

$$\begin{array}{ccc}\hfill e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,y\right)+d\left(z,w\right)\right)}& \le & max\left\{e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,z\right)+d\left(y,w\right)\right)},e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,w\right)+d\left(y,z\right)\right)}\right\}\hfill \\ & \le & e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,z\right)+d\left(y,w\right)\right)}+e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,w\right)+d\left(y,z\right)\right)}\hfill \end{array}$$

and the necessity is proved.

In order to prove the sufficiency, we assume that there exists arbitrarly large $\epsilon >0$ such that for all $x$, y, z, $w\in X$,

$$e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,y\right)+d\left(z,w\right)\right)}\le e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,z\right)+d\left(y,w\right)\right)}+e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,w\right)+d\left(y,z\right)\right)}.$$

But $e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,z\right)+d\left(y,w\right)\right)}+e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,w\right)+d\left(y,z\right)\right)}\le 2max\left\{e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,z\right)+d\left(y,w\right)\right)},e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,w\right)+d\left(y,z\right)\right)}\right\}$, hence

$$e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,z\right)+d\left(y,w\right)\right)}+e^{{\displaystyle \frac{\epsilon }{2}}\left(d\left(x,w\right)+d\left(y,z\right)\right)}\le e^{ln2+{\displaystyle \frac{\epsilon }{2}}max\left\{d\left(x,z\right)+d\left(y,w\right),d\left(x,w\right)+d\left(y,z\right)\right\}}.$$

It follows that ${\displaystyle \frac{\epsilon }{2}}\left(d\left(x,y\right)+d\left(z,w\right)\right)\le ln2+{\displaystyle \frac{\epsilon }{2}}max\left\{d\left(x,z\right)+d\left(y,w\right),d\left(x,w\right)+d\left(y,z\right)\right\}$. Dividing by ${\displaystyle \frac{\epsilon }{2}}$ and letting $\epsilon$ tend to infinity, as we may, we obtain (5). □

Let $\left(X,d\right)$ be a metric space and $M\subset X$ be a nonempty proper subset. Let $F:X\setminus M\to (0,\infty )$ be a function. We consider the following generalization of Barbilian’s logarithmic oscillation, defined for all $x,y\in X\setminus M$ by

$$d_{B,F}\left(x,y\right)=log{\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}.$$

Note that $d_{B,F}\left(x,y\right)=logmax\left\{{\displaystyle \frac{F\left(x\right)}{F\left(y\right)}},{\displaystyle \frac{F\left(y\right)}{F\left(x\right)}}\right\}=\left|log{\displaystyle \frac{F\left(x\right)}{F\left(y\right)}}\right|$.

Lemma 6.  (1) $d_{B,F}$ is a semi-distance on $X\setminus M$. Moreover, $d_{B,F}$ is a metric on $X\setminus M$ if and only if F is injective.

(2) $d_{B,F}$ is Gromov hyperbolic with $\delta =0$ on $X\setminus M$, therefore $d_{B,F}$ is strongly hyperbolic on $X\setminus M$ with an arbitrary parameter $\epsilon >0$.

Proof. (1) The non-negativity and symmetry of $d_{B,F}$ are obvious. Clearly, $x=y$ implies $d_{B,F}(x,y)=0$, but the converse holds if and only if F is injective. The triangle inequality $d_{B,F}\left(x,y\right)\le d_{B,F}\left(x,z\right)+d_{B,F}\left(z,y\right)$ follows from Lemma 3, letting $\alpha =F\left(x\right)$, $\beta =F\left(y\right)$ and $\gamma =F\left(z\right)$, and taking logarithms of both sides.

(2) The four-point inequality with $\delta =0$, namely

$$d_{B,F}\left(x,y\right)+d_{B,F}\left(z,w\right)\le max\left\{d_{B,F}\left(x,z\right)+d_{B,F}\left(y,w\right),d_{B,F}\left(x,w\right)+d_{B,F}\left(y,z\right)\right\},$$

follows from Lemma 3, letting $\alpha =F\left(x\right)$, $\beta =F\left(y\right)$, $\gamma =F\left(z\right)$ and $\omega =F\left(w\right)$, then taking logarithms of both sides. □

If $X={\mathbb{R}}^n$, $M=\left\{0_{{\mathbb{R}}^n}\right\}$ and $F\left(x\right)={\delta }_M\left(x\right)=\mathrm{dist}\left(x,M\right)$ for all $x\in X\setminus M$, then $d_{B,F}$

Theorem 1.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Let $F:X\setminus M\to (0,\infty )$ be a function. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive. Then $d_{M,F,f}$ defined by (1) is a metric on $X\setminus M$.

Proof. 

It is clear that $d_{M,F,f}(x,y)$ is well defined for all $x,y\in X\setminus M$ and that $d_{M,F,f}$$(x,y)=d_{M,F,f}(y,x)$ for all $x,y\in X\setminus M$. Also, $d_{M,F,f}(x,y)$ if and only if $x=y$, since f vanishes only at the origin.

We check the triangle inequality $d_{M,F,f}(x,y)\le d_{M,F,f}(x,z)+d_{M,F,f}(z,y)$, which is equivalent to

$${\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}+f\left(d(x,y)\right)\le \left({\displaystyle \frac{max\left\{F\left(x\right),F\left(z\right)\right\}}{min\left\{F\left(x\right),F\left(z\right)\right\}}}+f\left(d(x,z)\right)\right)\left({\displaystyle \frac{max\left\{F\left(z\right),F\left(y\right)\right\}}{min\left\{F\left(z\right),F\left(y\right)\right\}}}+f\left(d(z,y)\right)\right).$$

(6)

From (4) it follows that

$${\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\le {\displaystyle \frac{max\left\{F\left(x\right),F\left(z\right)\right\}}{min\left\{F\left(x\right),F\left(z\right)\right\}}}{\displaystyle \frac{max\left\{F\left(z\right),F\left(y\right)\right\}}{min\left\{F\left(z\right),F\left(y\right)\right\}}}.$$

(7)

By the triangle inequality for the metric d, the inequality ${\displaystyle \frac{max\left\{a,b\right\}}{min\left\{a,b\right\}}}\ge 1$ for $a,b>0$ and the assumption that f is nondecreasing and subadditive, we get

$$\begin{array}{ccc}\hfill f\left(d(x,y)\right)& \le & f\left(d(x,z)+d(z,y\right))\le f\left(d(x,z)\right)+f\left(d(z,y\right))\hfill \\ & \le & {\displaystyle \frac{max\left\{F\left(y\right),F\left(z\right)\right\}}{min\left\{F\left(y\right),F\left(z\right)\right\}}}f\left(d(x,z)\right)+{\displaystyle \frac{max\left\{F\left(x\right),F\left(z\right)\right\}}{min\left\{F\left(x\right),F\left(z\right)\right\}}}f\left(d(z,y)\right).\hfill \end{array}$$

(8)

Adding inequalities (7), (8) and $0\le f\left(d(x,z)\right)f\left(d(z,y)\right)$, we obtain (6). □

Remark 2.  

It suffices to prove Theorem 1 in the special case where f is the identity function of $[0,\infty )$. Then the general case follows, by replacing the metric d by the metric $f\circ d$.

We prove some properties of the metric $d_{M,F,f}$ , in the spirit of Ibragimov’s Theorem 2.1. (1), (3) and (4) in [17].

Theorem 2.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Let $F:X\setminus M\to (0,\infty )$ be a function. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive. Let $d_{M,F,f}(x,y)=log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}+f\left(d(x,y)\right)\right)$, $x,y\in X\setminus M$. Then the following assertions hold.

(i) For all $x,y\in X\setminus M$, $d_{M,F,f}(x,y)\ge log\left(1+f\left(d\right(x,y\left)\right)\right)$ and $d_{M,F,f}\left(x,y\right)\ge \left|log{\displaystyle \frac{F\left(x\right)}{F\left(y\right)}}\right|$;

(ii) The identity map $1_{X\setminus M}:\left(X\setminus M,d\right)\to \left(X\setminus M,d_{M,F,f}\right)$ is open.

(iii) If F is continuous on $X\setminus M$ and f is continuous at the origin, then the identity map $g:=1_{X\setminus M}:\left(X\setminus M,d\right)\to \left(X\setminus M,d_{M,F,f}\right)$ is a homeomorphism. Moreover, if F is $L-$Lipschitz for some $L>0$ and there exists $f^{\prime }\left(0\right)\in \mathbb{R}$, then the linear dilatation of g satisfies the inequality $H_g\left(x\right)\le 1+{\displaystyle \frac{L}{f^{\prime }\left(0\right)F\left(x\right)}}$, for all $x\in X\setminus M$.

(iv) Assume that the metric space $\left(X,d\right)$ is complete. If F has a continuous extension $\tilde{F}$ to $\left(X,d\right)$ with $\tilde{F}\left(x\right)=0$ for every $x\in M$ and f is a bijection continuous at the origin, then $\left(X\setminus M,d_{M,F,f}\right)$ is also a complete metric space.

Proof. (i) The inequalities follow from ${\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\ge 1$, $f\left(d\right(x,y\left)\right)\ge 0$ and

$log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\right)=\left|log{\displaystyle \frac{F\left(x\right)}{F\left(y\right)}}\right|$.

(ii) We have to prove that $1_{X\setminus M}:\left(X\setminus M,d_{M,F,f}\right)\to \left(X\setminus M,d\right)$ is continuous, i.e. for every sequence ${\left(x_n\right)}_{n\ge 1}$ in $X\setminus M$ and every $x\in X\setminus M$, $\underset{n\to \infty }{lim}{d}_{M,F,f}\left(x_n,x\right)=0$ implies $\underset{n\to \infty }{lim}d\left(x_n,x\right)=0$. Using the first inequality in (i) we see that $\underset{n\to \infty }{lim}{d}_{M,F,f}\left(x_n,x\right)=0$ implies $\underset{n\to \infty }{lim}f\left(d\left(x_n,x\right)\right)=0$, as $f\ge 0$. The claim follows using Lemma 1 (a).

(iii) By (ii), it remains to prove that $1_{X\setminus M}:\left(X\setminus M,d\right)\to \left(X\setminus M,d_{M,F,f}\right)$ is continuous, under the additional assumptions on F and f. Assume that $\underset{n\to \infty }{lim}d\left(x_n,x\right)=0$, where $x\in X\setminus M$ and $x_n\in X\setminus M$ for every $n\ge 1$. We have to prove that $\underset{n\to \infty }{lim}{d}_{M,F,f}\left(x_n,x\right)=0$. As F is continuous on $X\setminus M$, we get $\underset{n\to \infty }{lim}F\left(x_n\right)=F\left(x\right)>0$, hence $\underset{n\to \infty }{lim}{\displaystyle \frac{max\left\{F\left(x_n\right),F\left(x\right)\right\}}{min\left\{F\left(x_n\right),F\left(x\right)\right\}}}=1$. As f is continuous at the origin, $\underset{n\to \infty }{lim}f\left(d(x_n,x)\right)=0$. Then $\underset{n\to \infty }{lim}{d}_{M,F,f}\left(x_n,x\right)=0$.

Now assume that F is $L-$Lipschitz for some $L>0$ and there exists $f^{\prime }\left(0\right)\in \mathbb{R}$. Since f is nondecreasing, $f^{\prime }\left(0\right)\ge 0$. If $f^{\prime }\left(0\right)=0$, then by Lemma 1 (b) and $f\ge 0$ it follows that f is the constant null function, which contradicts the assumption that f vanishes only at the origin. Therefore, $f^{\prime }\left(0\right)>0$.

The linear dilatation of $g=1_{X\setminus M}$ at $x\in X\setminus M$ is $H_g\left(x\right)=\underset{r\to 0}{limsup}{H}_g\left(x,r\right)whereforall$$r>0$ we have in this special case

$$H_g\left(x,r\right)={\displaystyle \frac{sup\left\{d_{M,F,f}(x,y):d\left(x,y\right)\le r\right\}}{inf\left\{d_{M,F,f}(x,y):d\left(x,y\right)\ge r\right\}}}.$$

Let $r>0$. Since ${\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\ge 1$ and f is nondecreasing, it follows that

$$inf\left\{d_{M,F,f}(x,y):d\left(x,y\right)\ge r\right\}\ge log\left(1+f\left(r\right)\right)>0.$$

(9)

Note that ${\displaystyle \frac{max\left\{a,b\right\}}{min\left\{a,b\right\}}}=max\left\{{\displaystyle \frac{a}{b}},{\displaystyle \frac{b}{a}}\right\}=1+{\displaystyle \frac{\left|a-b\right|}{min\left\{a,b\right\}}}$, for every $a,b>0$.

Since $F:\left(X\setminus M,d\right)\to (0,\infty )$ is $L-$Lipschitz, we have ${\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}=1+{\displaystyle \frac{\left|F\left(x\right)-F\left(y\right)\right|}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\le 1+{\displaystyle \frac{Ld(x,y)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}$ and $min\left\{F\left(x\right),F\left(y\right)\right\}\ge F\left(x\right)-Ld(x,y)$. For $d(x,y)<{\displaystyle \frac{1}{L}}F\left(x\right)$, it follows that

$$d_{M,F,f}(x,y)\le log\left({\displaystyle \frac{F\left(x\right)}{F\left(x\right)-Ld(x,y)}}+f\left(d(x,y)\right)\right).$$

If $r<{\displaystyle \frac{1}{L}}F\left(x\right)$, using the fact that f is nondecreasing on $[0,\infty )$, the above estimate implies

$$sup\left\{d_{M,F,f}(x,y):d\left(x,y\right)\le r\right\}\le log\left({\displaystyle \frac{F\left(x\right)}{F\left(x\right)-Lr}}+f\left(r\right)\right).$$

(10)

From (9) and (10) we obtain

$$H_g\left(x,r\right)\le {\displaystyle \frac{log\left(\frac{F\left(x\right)}{F\left(x\right)-Lr}+f\left(r\right)\right)}{log\left(1+f\left(r\right)\right)}}$$

for every $r\in \left(0,{\displaystyle \frac{1}{L}}F\left(x\right)\right)$. It follows that

$$H_g\left(x\right)=\underset{r\to 0}{limsup}{H}_g\left(x,r\right)\le \underset{r\to 0}{limsup}{\displaystyle \frac{log\left(\frac{F\left(x\right)}{F\left(x\right)-Lr}+f\left(r\right)\right)}{log\left(1+f(r\right))}}.$$

But

$$\begin{array}{ccc}\hfill \underset{r\to 0}{limsup}{\displaystyle \frac{log\left(\frac{F\left(x\right)}{F\left(x\right)-Lr}+f\left(r\right)\right)}{log\left(1+f(r\right))}}& =& \underset{r\to 0}{lim}{\displaystyle \frac{log\left(\frac{F\left(x\right)}{F\left(x\right)-Lr}+f\left(r\right)\right)}{log\left(1+f(r\right))}}\hfill \\ & =& \underset{r\to 0}{lim}{\displaystyle \frac{\frac{Lr}{F\left(x\right)-Lr}+f\left(r\right)}{f\left(r\right)}}=1+{\displaystyle \frac{L}{F\left(x\right)}}\underset{r\to 0}{lim}{\displaystyle \frac{r}{f\left(r\right)}}=1+{\displaystyle \frac{L}{f^{\prime }\left(0\right)F\left(x\right)}}.\hfill \end{array}$$

(iv) Note that the nondecreasing function $f:[0,\infty )\to [0,\infty )$ is assumed here to be a bijection, therefore, it is increasing, hence its inverse $f^{-1}:[0,\infty )\to [0,\infty )$ is also increasing.

We prove that the metric space $\left(X\setminus M,d_{M,F,f}\right)$ is complete. Let ${\left(z_n\right)}_{n\ge 1}$ be a Cauchy sequence in $\left(X\setminus M,d_{M,F,f}\right)$. The first inequality in (i) implies that

$$d\left(x,y\right)\le f^{-1}(e^{d_{M,F,f}\left(x,y\right)}-1).$$

From the latter inequality, we deduce that ${\left(z_n\right)}_{n\ge 1}$ is also a Cauchy sequence in $\left(X,d\right)$, hence there exists $z\in X$ such that $\underset{n\to \infty }{lim}d\left(z_n,z\right)=0$.

Using the second inequality in (i) and the assumption that ${\left(z_n\right)}_{n\ge 1}$ is a Cauchy sequence in $\left(X\setminus M,d_{M,F,f}\right)$, it follows that ${\left(logF\left(z_n\right)\right)}_{n\ge 1}$ is a Cauchy sequence in $\mathbb{R}$. In particular, ${\left(logF\left(z_n\right)\right)}_{n\ge 1}$ is bounded. Denote $\alpha =\underset{n\ge 1}{inf}$$logF\left(z_n\right)\in \mathbb{R}$ and $\beta =\underset{n\ge 1}{sup}$$logF\left(z_n\right)\in \mathbb{R}$. Then $e^{\alpha }=\underset{n\ge 1}{inf}$$F\left(z_n\right)\le \underset{n\ge 1}{sup}F\left(z_n\right)=e^{\beta }$. Since $\tilde{F}$ is continuous on $(X,d)$, $\tilde{F}\left(z\right)=\underset{n\to \infty }{lim}\tilde{F}\left(z_n\right)=\underset{n\to \infty }{lim}F\left(z_n\right)\ge e^{\alpha }>0$. As $\tilde{F}$ vanishes on M, we get $z\in X\setminus M$, hence $F\left(z\right)=\tilde{F}\left(z\right)=\underset{n\to \infty }{lim}F\left(z_n\right)$. Then $\underset{n\to \infty }{lim}{\displaystyle \frac{max\left\{F\left(z_n\right),F\left(z\right)\right\}}{min\left\{F\left(z_n\right),F\left(z\right)\right\}}}=1.$

Finally, since f is continuous at the origin, it follows that

$\underset{n\to \infty }{lim}{d}_{M,F,f}\left(z_n,z\right)=\underset{n\to \infty }{lim}log\left({\displaystyle \frac{max\left\{F\left(z_n\right),F\left(z\right)\right\}}{min\left\{F\left(z_n\right),F\left(z\right)\right\}}}+f\left(d(z_n,z)\right)\right)=0$, hence, ${\left(z_n\right)}_{n\ge 1}$ converges to z in $\left(X\setminus M,d_{M,F,f}\right)$. □

4. Gromov Hyperbolicity of a Stabilizing Metric in a Ptolemaic Metric Space

In [6], Theorem 4.2 it is proved that, for the punctured open unit ball $G={\mathbb{B}}^n\setminus \left\{0_{{\mathbb{R}}^n}\right\}$ and $M=\left\{0_{{\mathbb{R}}^n}\right\}$, the metric $d_{G,M}$ is Gromov hyperbolic with constant $\delta \le log3$. We will generalize this result in the setting of a Ptolemaic metric space $\left(X,d\right)$, considering M as a singleton, $M=\left\{p\right\}$, and $G=X\setminus M$ and obtaining a smaller Gromov hyperbolicity constant, that is, $\delta \le log2.$

Lemma 7.  

For all $a,b,c\in (0,\infty )$ the following inequality holds

$$amax\left\{{\displaystyle \frac{b}{c}},{\displaystyle \frac{c}{b}}\right\}\le max\left\{bmax\left\{{\displaystyle \frac{c}{a}},{\displaystyle \frac{a}{c}}\right\},cmax\left\{{\displaystyle \frac{a}{b}},{\displaystyle \frac{b}{a}}\right\}\right\}.$$

In particular,

$$amax\left\{{\displaystyle \frac{b}{c}},{\displaystyle \frac{c}{b}}\right\}\le bmax\left\{{\displaystyle \frac{c}{a}},{\displaystyle \frac{a}{c}}\right\}+cmax\left\{{\displaystyle \frac{a}{b}},{\displaystyle \frac{b}{a}}\right\}.$$

(11)

Proof. 

Due to symmetry, it suffices to prove the first inequality under the assumption $b\le c$. The claim (11) will be rewritten under the following forms, depending on the position of a with respect to the interval $\left[b,c\right]$.

Case 1. If $a\le b\le c$: ${\displaystyle \frac{ac}{b}}\le {\displaystyle \frac{bc}{a}}$, that is equivalent to ${\displaystyle \frac{c}{ab}}(a-b)(a+b)\le 0$;

Case 2. If $b\le a\le c$: ${\displaystyle \frac{ac}{b}}\le max\left\{{\displaystyle \frac{bc}{a}},{\displaystyle \frac{ac}{b}}\right\}$;

Case 3. If $b\le c\le a$: ${\displaystyle \frac{ac}{b}}\le max\left\{{\displaystyle \frac{ba}{c}},{\displaystyle \frac{ac}{b}}\right\}$.

So, the claim is proved in all cases. □

Theorem 3.  

Let $\left(X,d\right)$ be a metric space. Given $p\in X$, denote $\delta \left(x\right)=d(x,p),x\in X$. Define $d_p\left(x,y\right)=log\left({\displaystyle \frac{max\left\{\delta \left(x\right),\delta \left(y\right)\right\}}{min\left\{\delta \left(x\right),\delta \left(y\right)\right\}}}+d(x,y)\right)$ for all $x,y\in X\setminus \left\{p\right\}$. If $\left(X,d\right)$ is Ptolemaic, then $\left(X\setminus \left\{p\right\},d_p\right)$ is Gromov hyperbolic with constant $\delta \le log2.$

Proof. 

We look for $\delta \ge 0$ such that the following four-point condition holds for all x, y, z, $w\in X\setminus \left\{p\right\}$

$$d_p\left(x,y\right)+d_p\left(z,w\right)\le max\left\{d_p\left(x,z\right)+d_p\left(y,w\right),d_p\left(x,w\right)+d_p\left(y,z\right)\right\}+2\delta .$$

(12)

For brevity, denote $d(x,y)=\left|xy\right|$, $\delta \left(x\right)={\delta }_x$ etc. Note that ${\displaystyle \frac{max\left\{a,b\right\}}{min\left\{a,b\right\}}}=max\left\{{\displaystyle \frac{a}{b}},{\displaystyle \frac{b}{a}}\right\}$ for all positive numbers a and b.

We may assume that ${\delta }_x\le min\left\{{\delta }_y,{\delta }_z,{\delta }_w\right\}$, due to the symmetry of (12) with respect to the variables x, y, z and w.

Then

$$\begin{array}{ccc}\hfill e^{d_p\left(x,y\right)+d_p\left(z,w\right)}& =& \left|xy\right|\left|zw\right|+{\displaystyle \frac{{\delta }_y}{{\delta }_x}}max\left\{{\displaystyle \frac{{\delta }_z}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_z}}\right\}\hfill \\ & & +\left|xy\right|max\left\{{\displaystyle \frac{{\delta }_z}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_z}}\right\}+\left|zw\right|{\displaystyle \frac{{\delta }_y}{{\delta }_x}}.\hfill \end{array}$$

and

$$\begin{array}{ccc}& & e^{d_p\left(x,z\right)+b_p\left(y,w\right)}+e^{d_p\left(x,w\right)+b_p\left(y,z\right)}\hfill \\ & =& \left(\left|xz\right|\left|yw\right|+\left|xw\right|\left|yz\right|\right)+\left({\displaystyle \frac{{\delta }_z}{{\delta }_x}}max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_y}}\right\}+{\displaystyle \frac{{\delta }_w}{{\delta }_x}}max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_z}},{\displaystyle \frac{{\delta }_z}{{\delta }_y}}\right\}\right)\hfill \\ & & +\left(\left|yz\right|{\displaystyle \frac{{\delta }_w}{{\delta }_x}}+\left|yw\right|{\displaystyle \frac{{\delta }_z}{{\delta }_x}}\right)+\left(\left|xz\right|max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_y}}\right\}+\left|xw\right|max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_z}},{\displaystyle \frac{{\delta }_z}{{\delta }_y}}\right\}\right).\hfill \end{array}$$

Taking $a={\delta }_y$, $b={\delta }_z$ and $c={\delta }_w$ in (11), then dividing by ${\delta }_x$, we obtain

$${\displaystyle \frac{{\delta }_y}{{\delta }_x}}max\left\{{\displaystyle \frac{{\delta }_z}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_z}}\right\}\le {\displaystyle \frac{{\delta }_z}{{\delta }_x}}max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_y}}\right\}+{\displaystyle \frac{{\delta }_w}{{\delta }_x}}max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_z}},{\displaystyle \frac{{\delta }_z}{{\delta }_y}}\right\}.$$

(13)

Apply Ptolemy’s inequality in $\left(X,d\right)$ several times, as follows.

1) For $\left(x,z,y,w\right)$,

$$\left|xy\right|\left|zw\right|\le \left|xz\right|\left|yw\right|+\left|xw\right|\left|yz\right|.$$

(14)

2) For $\left(z,y,w,p\right)$,

$$\left|zw\right|{\displaystyle \frac{{\delta }_y}{{\delta }_x}}\le \left|yz\right|{\displaystyle \frac{{\delta }_w}{{\delta }_x}}+\left|yw\right|{\displaystyle \frac{{\delta }_z}{{\delta }_x}}.$$

(15)

3) For $\left(x,z,y,p\right)$,

$$\left|xy\right|{\displaystyle \frac{{\delta }_z}{{\delta }_w}}\le \left|xz\right|{\displaystyle \frac{{\delta }_y}{{\delta }_w}}+\left|yz\right|{\displaystyle \frac{{\delta }_x}{{\delta }_w}}$$

(16)

4) For $\left(x,w,y,p\right)$,

$$\left|xy\right|{\displaystyle \frac{{\delta }_w}{{\delta }_z}}\le \left|xw\right|{\displaystyle \frac{{\delta }_y}{{\delta }_z}}+\left|yw\right|{\displaystyle \frac{{\delta }_x}{{\delta }_z}}$$

(17)

Now (16) and (17) together imply

$$\left|xy\right|max\left\{{\displaystyle \frac{{\delta }_z}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_z}}\right\}\le max\left\{\left|xz\right|{\displaystyle \frac{{\delta }_y}{{\delta }_w}}+\left|yz\right|{\displaystyle \frac{{\delta }_x}{{\delta }_w}},\left|xw\right|{\displaystyle \frac{{\delta }_y}{{\delta }_z}}+\left|yw\right|{\displaystyle \frac{{\delta }_x}{{\delta }_z}}\right\}$$

(18)

Adding (13), (14) and (15) we obtain

$$\begin{array}{ccc}& & e^{d_p\left(x,y\right)+d_p\left(z,w\right)}-\left|xy\right|max\left\{{\displaystyle \frac{{\delta }_z}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_z}}\right\}\hfill \\ & \le & e^{d_p\left(x,z\right)+b_p\left(y,w\right)}+e^{d_p\left(x,w\right)+b_p\left(y,z\right)}-\left(\left|xz\right|max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_y}}\right\}+\left|xw\right|max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_z}},{\displaystyle \frac{{\delta }_z}{{\delta }_y}}\right\}\right)\hfill \end{array}$$

But (18) yields

$$\begin{array}{ccc}& & \left|xy\right|max\left\{{\displaystyle \frac{{\delta }_z}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_z}}\right\}\le \left(\left|xz\right|max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_w}},{\displaystyle \frac{{\delta }_w}{{\delta }_y}}\right\}+\left|xw\right|max\left\{{\displaystyle \frac{{\delta }_y}{{\delta }_z}},{\displaystyle \frac{{\delta }_z}{{\delta }_y}}\right\}\right)\hfill \\ & & +\left(\left|yz\right|{\displaystyle \frac{{\delta }_x}{{\delta }_w}}+\left|xw\right|{\displaystyle \frac{{\delta }_y}{{\delta }_z}}\right).\hfill \end{array}$$

The latter two inequalities imply

$$\begin{array}{ccc}\hfill e^{d_p\left(x,y\right)+d_p\left(z,w\right)}& \le & e^{d_p\left(x,z\right)+d_p\left(y,w\right)}+e^{d_p\left(x,w\right)+d_p\left(y,z\right)}+\left(\left|yz\right|{\displaystyle \frac{{\delta }_x}{{\delta }_w}}+\left|xw\right|{\displaystyle \frac{{\delta }_y}{{\delta }_z}}\right)\hfill \\ & \le & 2\left(e^{d_p\left(x,z\right)+d_p\left(y,w\right)}+e^{d_p\left(x,w\right)+d_p\left(y,z\right)}\right)\le 4max\left\{e^{d_p\left(x,z\right)+d_p\left(y,w\right)},e^{d_p\left(x,w\right)+d_p\left(y,z\right)}\right\}\hfill \end{array}$$

and (12) follows with $\delta =log2$. □

Without assuming that $\left(X,d\right)$ is Ptolemaic, we may prove that for a modified stabilizing metric $d_{M,F,f}$ with $M=\left\{p\right\}$, $F\left(x\right)=d(x,p),x\in X$ and f as in the statement of Theorem 1, the restriction of $d_{M,F,f}$ to every bounded set $G\subset X\setminus \left\{p\right\}$ is Gromov hyperbolic if there exists $\lambda >0$ such that $f\left(t\right)\le \lambda t$ for every $t\in [0,\infty )$. We will use the comparison between this restriction of $d_{M,F,f}$ and the Barbilian logarithmic oscillation, as well as the invariance of Gromov hyperbolicity under rough isometries.

It is known that, if there exists a rough isometry $\mathrm{\Phi }:(X,d_X)\to (Y,d_Y)$ between two complete geodesic metric spaces and if $(X,d_X)$ is Gromov hyperbolic, then $(Y,d_Y)$ is Gromov hyperbolic [22]. We will show that this invariance property holds for rough isometries between semi-metric spaces, which are not necessarily geodesic.

Definition 1.  

Let $\alpha ,\beta ,\gamma$ be nonnegative numbers. We say that a map between semi-metric spaces $\mathrm{\Phi }:(X,d_X)\to (Y,d_Y)$ is

(1) an $\left(\alpha ,\beta \right)-$rough isometry if Φ satisfies the distortion condition

$$d_X(x_1,x_2)-\alpha \le d_Y(\mathrm{\Phi }\left(x_1\right),\mathrm{\Phi }\left(x_2\right))\le d_X(x_1,x_2)+\beta$$

(19)

for all $x_1,x_2\in X$;

(2) $\gamma -$roughly surjective if for every $y\in Y$ there exists $x_y$ such that $d_Y(y,\mathrm{\Phi }\left(x_y\right))\le \gamma$.

In [22], Definition 2.6 a $\tau -$rough isometry between metric spaces is defined as an $\left(\alpha ,\beta \right)-$rough isometry that is $\gamma -$roughly surjective, with $\alpha =\beta =\gamma =:\tau$. A map $\mathrm{\Phi }:(X,d_X)\to (Y,d_Y)$ is said to be a rough isometry if it is a $\tau -$rough isometry for some $\tau \ge 0$. Conversely, every $\left(\alpha ,\beta \right)-$rough isometry between metric spaces, that is $\gamma -$roughly surjective, is a $\tau -$rough isometry with $\tau =max\left\{\alpha ,\beta ,\gamma \right\}$.

Lemma 8.  

Assume that $\mathrm{\Phi }:(X,d_X)\to (Y,d_Y)$ is a $\gamma -$roughly surjective rough $\left(\alpha ,\beta \right)-$isometry between semi-metric spaces. If $(X,d_X)$ is Gromov hyperbolic with a constant $\delta \ge 0$, that $(Y,d_Y)$ is Gromov hyperbolic with a constant ${\delta }^{\prime }\le \delta +\alpha +\beta +4\gamma$.

Proof. 

Let $x_1,x_2,x_3,x_4\in X$. By the second inequality in (19),

$$d_Y(\mathrm{\Phi }\left(x_1\right),\mathrm{\Phi }\left(x_2\right))+d_Y(\mathrm{\Phi }\left(x_3\right),\mathrm{\Phi }\left(x_4\right))\le d_X(x_1,x_2)+d_X(x_3,x_4)+2\beta .$$

As $(X,d_X)$ is Gromov hyperbolic with a constant $\delta \ge 0$,

$$d_X(x_1,x_2)+d_X(x_3,x_4)\le max\left\{d_X(x_1,x_3)+d_X(x_2,x_4),d_X(x_1,x_4)+d_X(x_2,x_3)\right\}+2\delta .$$

By the first inequality in (19),

$$\begin{array}{ccc}& & max\left\{d_X(x_1,x_3)+d_X(x_2,x_4),d_X(x_1,x_4)+d_X(x_2,x_3)\right\}-2\alpha \hfill \\ & \le & max\left\{d_X(\mathrm{\Phi }\left(x_1\right),\mathrm{\Phi }\left(x_3\right))+d_X(\mathrm{\Phi }\left(x_2\right),\mathrm{\Phi }\left(x_4\right)),d_X(\mathrm{\Phi }\left(x_1\right),\mathrm{\Phi }\left(x_4\right))+d_X(\mathrm{\Phi }\left(x_2\right),\mathrm{\Phi }\left(x_3\right))\right\}.\hfill \end{array}$$

The latter three inequalities imply

$$\begin{array}{ccc}& & d_Y(\mathrm{\Phi }\left(x_1\right),\mathrm{\Phi }\left(x_2\right))+d_Y(\mathrm{\Phi }\left(x_3\right),\mathrm{\Phi }\left(x_4\right))-2(\alpha +\beta +\delta )\hfill \\ & \le & max\left\{d_X(\mathrm{\Phi }\left(x_1\right),\mathrm{\Phi }\left(x_3\right))+d_X(\mathrm{\Phi }\left(x_2\right),\mathrm{\Phi }\left(x_4\right)),d_X(\mathrm{\Phi }\left(x_1\right),\mathrm{\Phi }\left(x_4\right))+d_X(\mathrm{\Phi }\left(x_2\right),\mathrm{\Phi }\left(x_3\right))\right\}.\hfill \end{array}$$

(20)

The claim follows in the case where $\gamma =0$, i. e. $\mathrm{\Phi }$ is surjective. In the general case, due to the coarse surjectivity of $\mathrm{\Phi }$ we find for every $y_1,y_2,y_3,y_4\in Y$ some points $x_1,x_2,x_3,x_4\in X$ such that $d_Y(y_k,\mathrm{\Phi }\left(x_k\right))\le \gamma$ for all $k\in \left\{1,2,3,4\right\}$. Denote ${\delta }_{ij}=d_Y(y_i,y_j)$ and ${\Delta }_{ij}=d_Y(\mathrm{\Phi }\left(x_i\right),\mathrm{\Phi }\left(x_j\right))$ for $i,j\in \left\{1,2,3,4\right\}$. Then $\left|{\delta }_{ij}-{\Delta }_{ij}\right|\le 2\gamma$ for all $i,j\in \left\{1,2,3,4\right\}$.

Finally, taking into account (20) we get

$$\begin{array}{ccc}\hfill {\delta }_{12}+{\delta }_{34}& \le & {\Delta }_{12}+{\Delta }_{34}+4\gamma \le max\left\{{\Delta }_{13}+{\Delta }_{24},{\Delta }_{14}+{\Delta }_{23}\right\}+2(\alpha +\beta +\delta )+4\gamma \hfill \\ & \le & max\left\{{\delta }_{13}+{\delta }_{24},{\delta }_{14}+{\delta }_{23}\right\}+2(\alpha +\beta +\delta )+8\gamma .\hfill \end{array}$$

□

Remark 3.  

In [31], Theorem 3.20, Väisälä proved the following result. If there exists a map between intrinsic metric spaces $\mathrm{\Phi }:(X,d_X)\to (Y,d_Y)$, that is a $\mu -$roughly surjective $\left(\lambda ,\mu \right)-$quasi-isometry and if X is $\delta -$hyperbolic, then Y is ${\delta }^{\prime }-$hyperbolic for some ${\delta }^{\prime }={\delta }^{\prime }(\delta ,\lambda ,\mu )$. $\mathrm{\Phi }:(X,d_X)\to (Y,d_Y)$ is said to be a $\left(\lambda ,\mu \right)-$quasi-isometry if ${\lambda }^{-1}{d}_X(x_1,x_2)-\mu \le d_Y(\mathrm{\Phi }\left(x_1\right),\mathrm{\Phi }\left(x_2\right))\le \lambda d_X(x_1,x_2)+\mu$ for all $x_1,x_2\in X$. Note that a $\tau -$roughly surjective $\left(1,\tau \right)-$quasi-isometry is a $\tau -$rough isometry in the sense of [22], Definition 2.6.

Now we prove the Gromov hyperbolicity of a stabilizing metric in a bounded subdomain of a punctured space $X\setminus \left\{p\right\}$, for $F\left(x\right)=\mathrm{dist}(x,p)$, using the fact that this metric is roughly isometric to the Barbilian’s logarithmic oscillation.

Theorem 4.  

Let $\left(X,d\right)$ be a metric space. Given $p\in X$, denote $\delta \left(x\right)=d(x,p),x\in X$. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive, and that there exists $\lambda >0$ such that $f\left(t\right)\le \lambda t$ for every $t\in [0,\infty )$. Define $d_{p,f}(x,y)=log\left({\displaystyle \frac{max\left\{\delta \left(x\right),\delta \left(y\right)\right\}}{min\left\{\delta \left(x\right),\delta \left(y\right))\right\}}}+f\left(d(x,y)\right)\right)$, $x,y\in X\setminus \left\{p\right\}$. If $G\subset X\setminus \left\{p\right\}$ is bounded, then the restriction of $d_{p,f}$ to $G\times G$ is Gromov hyperbolic with $\delta \le log\left(1+2\lambda K\right)$, where $K=\underset{z\in G}{sup}d(z,p)\le \mathrm{dist}(p,G)+\mathrm{diam}\left(G\right).$

Proof. 

We follow ideas from the proof of [6], Lemma 4.1, Theorem 5.1.

For all $x,y\in X$ it follows from the triangle inequality that $d(x,y)\le \delta \left(x\right)+\delta \left(y\right)\le 2max\left\{\delta \left(x\right),\delta \left(y\right)\right\}$, hence $f\left(d(x,y)\right)\le 2\lambda max\left\{\delta \left(x\right),\delta \left(y\right)\right\}.$

For every $x\in G$ we have $d(x,p)\le K$, where $K=\underset{z\in G}{sup}d(z,p)\le \mathrm{dist}(p,G)+\mathrm{diam}\left(G\right)$. Then $min\left\{\delta \left(x\right),\delta \left(y\right))\right\}\le K$, hence $1\le {\displaystyle \frac{K}{min\left\{\delta \left(x\right),\delta \left(y\right))\right\}}}$. From the latter inequality and $f\left(d(x,y)\right)\le 2\lambda max\left\{\delta \left(x\right),\delta \left(y\right)\right\}$ we obtain $f\left(d(x,y)\right)\le 2\lambda K{\displaystyle \frac{max\left\{\delta \left(x\right),\delta \left(y\right)\right\}}{min\left\{\delta \left(x\right),\delta \left(y\right))\right\}}}$, for all $x,y\in G.$

Therefore, for all $x,y\in G$,

$$log\left({\displaystyle \frac{max\left\{\delta \left(x\right),\delta \left(y\right)\right\}}{min\left\{\delta \left(x\right),\delta \left(y\right))\right\}}}\right)\le d_{p,f}(x,y)\le log\left({\displaystyle \frac{max\left\{\delta \left(x\right),\delta \left(y\right)\right\}}{min\left\{\delta \left(x\right),\delta \left(y\right))\right\}}}\right)+log\left(1+2\lambda K\right).$$

As the Barbilian logarithmic oscillation $log\left({\displaystyle \frac{max\left\{\delta \left(x\right),\delta \left(y\right)\right\}}{min\left\{\delta \left(x\right),\delta \left(y\right))\right\}}}\right)$ is $0-$hyperbolic in the sense of Gromov on $X\setminus \left\{p\right\}$, in particular on G, taking $\mathrm{\Phi }=1_G$ and $\alpha =0$, $\beta =log\left(1+2\lambda K\right),\gamma =0$, $\delta =0$ in Lemma 8 it follows that the restriction of $d_{M,F,f}$ to G is Gromov hyperbolic with $\delta \le log\left(1+2\lambda K\right)$. □

Corollary 1.  

Let $\left(X,d\right)$ be a metric space. Given $p\in X$, denote $\delta \left(x\right)=d(x,p),x\in X$. Let $d_p(x,y)=log\left({\displaystyle \frac{max\left\{\delta \left(x\right),\delta \left(y\right)\right\}}{min\left\{\delta \left(x\right),\delta \left(y\right))\right\}}}+d(x,y)\right)$, $x,y\in X\setminus \left\{p\right\}$. If X is bounded, then $d_p$ is Gromov hyperbolic with $\delta \le log\left(1+2K\right)$, where $K=\underset{z\in X\setminus \left\{p\right\}}{sup}d(z,p)\le \mathrm{dist}(p,G)+\mathrm{diam}\left(G\right)$, $G=X\setminus \left\{p\right\}$

5. A New Generalization of Vuorinen’s Distance Ratio Metric

The distance ratio metric introduced by Vuorinen [32] is defined by $j_G\left(x,y\right)=log\left(1+{\displaystyle \frac{d(x,y)}{min\left\{d_G\left(x\right),d_G\left(y\right)\right\}}}\right)$, where $x,y\in G$ and $G\subset {\mathbb{R}}^n$ is an open set with non-empty boundary. Here $d_G\left(x\right)=\mathrm{dist}\left(x,\partial G\right)=inf\left\{d(x,y):y\in \partial G\right\}$.

For $G={\mathbb{R}}^n\setminus \left\{0\right\}$, assuming that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive, and in addition, that there exists $\lambda \ge 1$ such that $t\le f\left(t\right)\le \lambda t$ for all $t\in [0,\infty )$, Boskoff and Suceavă proved in [6], Theorem 6.3 that $j_{G,f}(x,y):=log\left(1+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{d_G\left(x\right),d_G\left(y\right)\right\}}}\right)$ defines a distance on ${\mathbb{R}}^n\setminus \left\{0\right\}$, that is Gromov hyperbolic with constant $\delta \le 2log(1+2\lambda )$. If f is the identity of $[0,\infty )$ it was proved by Hästö that $j_{G,f}$ is Gromov hyperbolic with constant $\delta \le log3$.

In [26] we studied a generalization of Vuorinen’s distance ratio metric in an arbitrary metric space $(X,d)$, which occurs by replacing the distance to a domain’s boundary by any $1-$Lipschitz function. Namely, we defined

$$j_F\left(x,y\right)=log\left(1+{\displaystyle \frac{d(x,y)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\right)$$

for $x,y\in X\setminus M$, where $F:\left(X\setminus M,d\right)\to (0,\infty )$ is a $1-$Lipschitz function. We proved that this is a metric and that many other metrics, studied in [25] as generalizations of well-known hyperbolic-type metrics, are bi-Lipschitz equivalent to $j_F$. Very recently, a similar metric with many applications has been considered in [24].

In order to generalize both $d_{V,f}$ and $j_F$ we define for all $x,y\in X\setminus M$

$$j_{F,f}\left(x,y\right)=log\left(1+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\right),$$

where $F:X\setminus M\to (0,\infty )$ and $f:[0,\infty )\to [0,\infty )$ are given functions.

Using the following result, which is Theorem 2.1. (1) in [26], we obtain a consequence that generalizes both this result and Theorem 6. 3 (a) of [6], Theorem 4.2.

Proposition 1.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Let $F:\left(X\setminus M,d\right)\to (0,\infty )$ be a $1-$Lipschitz function. Then the following properties hold:

(1) $j_F$ is a metric on $X\setminus M$.

(2) The identity map $1_{X\setminus M}:\left(X\setminus M,d\right)\to \left(X\setminus M,j_F\right)$ is $1-$quasiconformal;

(3) If F has a continuous extension $\tilde{F}$ to $\left(X,d\right)$ with $\tilde{F}\left(x\right)=0$ for every $x\in M$ and $\left(X,d\right)$ is complete, then $\left(X\setminus M,j_F\right)$ is also a complete metric space.

Theorem 5.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive. Let $F:X\setminus M\to (0,\infty )$ such that, for all $x,y\in X\setminus M$,

$$\left|F\left(x\right)-F\left(y\right)\right|\le f\left(d(x,y)\right).$$

(21)

Then $j_{F,f}$ is a metric on $X\setminus M$.

Proof. 

Define $D(x,y)=f\left(d\right(x,y\left)\right)$ for all $x,y\in X\setminus M$. Then D is a metric on $X\setminus M$. By (21), F is $1-$Lipschitz on $(X\setminus M,D)$. Applying Proposition 1 with d replaced by D, we obtain that $j_{F,f}$ is a metric on $X\setminus M$. □

Corollary 2.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Let $F:\left(X\setminus M,d\right)\to (0,\infty )$ be a $1-$Lipschitz function. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive, and that $f\left(t\right)\ge t$ for all $t\in [0,\infty )$. Then $j_{F,f}$ is a metric on $X\setminus M$.

Proof. 

For all $x,y\in X\setminus M$ we have $\left|F\left(x\right)-F\left(y\right)\right|\le d(x,y)\le f\left(d(x,y)\right)$, as F is $1-$Lipschitz on $(X\setminus M,d)$ and $f\left(t\right)\ge t$ for all $t\in [0,\infty )$. The assumptions of Theorem 5 are satisfied. □

Note that Proposition 1 is recovered from Corollary 2, in the case where f is the identity function.

We will give a direct proof for Theorem 5, in order to highlight the underlying mechanism of the proof of Proposition 1, which can also be applied in other contexts.

Lemma 9.  

Let $a,b,c,u,v,w\in (0,\infty )$ such that $\left|u-v\right|\le a$, $\left|v-w\right|\le b$ and $\left|w-u\right|\le c$. Assume that $a\le b+c$. Then

$$1+{\displaystyle \frac{a}{min\left\{u,v\right\}}}\le \left(1+{\displaystyle \frac{b}{min\left\{v,w\right\}}}\right)\left(1+{\displaystyle \frac{c}{min\left\{w,u\right\}}}\right).$$

(22)

Proof. 

As $a\le b+c$, it suffices to prove that

$${\displaystyle \frac{b+c}{min\left\{u,v\right\}}}\le {\displaystyle \frac{b}{min\left\{v,w\right\}}}+{\displaystyle \frac{c}{min\left\{w,u\right\}}}+{\displaystyle \frac{b}{min\left\{v,w\right\}}}{\displaystyle \frac{c}{min\left\{w,u\right\}}}.$$

(23)

Denote $E={\displaystyle \frac{b+c}{min\left\{u,v\right\}}}-{\displaystyle \frac{b}{min\left\{v,w\right\}}}-{\displaystyle \frac{c}{min\left\{w,u\right\}}}-{\displaystyle \frac{b}{min\left\{v,w\right\}}}{\displaystyle \frac{c}{min\left\{w,u\right\}}}$. Then (23) is equivalent to $E\le 0$.

Due to the symmetry of the above inequality with respect to the pair $\left(b,v\right)$ and $\left(c,u\right)$, we may assume that $u\le v$.

Case 1.If $w\le u\le v$, then $E={\displaystyle \frac{b+c}{u}}-{\displaystyle \frac{b}{w}}-{\displaystyle \frac{c}{w}}-{\displaystyle \frac{bc}{w^2}}{\displaystyle \frac{c}{min\left\{w,u\right\}}}=(b+c){\displaystyle \frac{w-u}{uw}}-{\displaystyle \frac{bc}{w^2}}$, hence $E\le 0$.

Case 2.If $u\le w\le v$, then $E={\displaystyle \frac{b+c}{u}}-{\displaystyle \frac{b}{w}}-{\displaystyle \frac{c}{u}}-{\displaystyle \frac{bc}{uw}}={\displaystyle \frac{b}{uw}}(w-u)-{\displaystyle \frac{bc}{uw}}$. Hence, $E={\displaystyle \frac{b}{uw}}(w-u-c)\le 0$, as $w-u\le \left|w-u\right|\le c.$

Case 3.If $u\le v\le w$, then $E={\displaystyle \frac{b+c}{u}}-{\displaystyle \frac{b}{v}}-{\displaystyle \frac{c}{u}}-{\displaystyle \frac{bc}{uv}}={\displaystyle \frac{b}{uv}}(v-u)-{\displaystyle \frac{bc}{uv}}$. It follows that $E={\displaystyle \frac{b}{uv}}(v-u-c)\le 0$, as $v-u\le w-u\le \left|w-u\right|\le c.$□

Now we provide a direct proof of Theorem 5.

Proof. 

Let $j_{F,f}\left(x,y\right)=log\left(1+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\right)$, $x,y\in X\setminus M$.

As F is positive on $X\setminus M$ and f is nonnegative and vanishes only at the origin, we see that $j_{F,f}\left(x,y\right)$ is a nonnegative real number and $j_{F,f}\left(x,y\right)=0$ if and only if $x=y$. Clearly, $j_{F,f}\left(x,y\right)=j_{F,f}\left(y,x\right)$ for all $x,y\in X\setminus M$.

We check that $j_f$ satisfies the triangle inequality $j_{F,f}\left(x,y\right)\le j_{F,f}\left(x,z\right)+j_{F,f}\left(z,y\right)$ for all $x,y,z\in X\setminus M$. This is equivalent to

$${\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\le {\displaystyle \frac{f\left(d\right(x,z\left)\right)}{min\left\{F\left(x\right),F\left(z\right)\right\}}}+{\displaystyle \frac{f\left(d\right(z,y\left)\right)}{min\left\{F\left(z\right),F\left(y\right)\right\}}}+{\displaystyle \frac{f\left(d\right(x,z\left)\right)}{min\left\{F\left(x\right),F\left(z\right)\right\}}}{\displaystyle \frac{f\left(d\right(z,y\left)\right)}{min\left\{F\left(z\right),F\left(y\right)\right\}}}.$$

(24)

Denote $a=f\left(d\right(x,y\left)\right)$, $b=f\left(d\right(y,z\left)\right)$, $c=f\left(d\right(z,x\left)\right)$ and $u=F\left(x\right)$, $v=F\left(y\right)$, $w=F\left(z\right)$.

As f is nondecreasing and subadditive, $a\le b+c$, by the triangle inequality for the metric d.

By (21) and its counterparts, it follows that $\left|u-v\right|\le a$, $\left|v-w\right|\le b$ and $\left|w-u\right|\le c$.

Using Lemma 9 we obtain the inequality (22), which in our case is equivalent to (24). The proof is completed. □

Lemma 10.  

Under the assumptions from Theorem 5 the metric $j_{F,f}$ satisfies the following inequalities, for all $x,y\in X\setminus M$:

$$j_{F,f}\left(x,y\right)\ge log\left(1+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{F\left(x\right)}}\right)$$

and

$$j_{F,f}\left(x,y\right)\ge \left|log{\displaystyle \frac{F\left(x\right)}{F\left(y\right)}}\right|.$$

Proof. 

The first inequality is obvious. Since $max\left\{F\left(x\right),F\left(y\right)\right\}-min\left\{F\left(x\right),F\left(y\right)\right\}=\left|F\left(x\right)-F\left(y\right)\right|\le f\left(d(x,y)\right)$, it follows that

$j_{F,f}\left(x,y\right)=log{\displaystyle \frac{min\left\{F\left(x\right),F\left(y\right)\right\}+f\left(d(x,y)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\ge log{\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}=\left|log{\displaystyle \frac{F\left(x\right)}{F\left(y\right)}}\right|.$□

Theorem 6.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive. Let $F:X\setminus M\to (0,\infty )$ such that $\left|F\left(x\right)-F\left(y\right)\right|\le f\left(d(x,y)\right)$ for all $x,y\in X\setminus M$. Consider the identity map $g:=1_{X\setminus M}:\left(X\setminus M,d\right)\to \left(X\setminus M,j_{F,f}\right).$ Then the following properties hold:

(1) g is open;

(2) If f is continuous at the origin, then g is an $1-$ quasiconformal homeomorphism;

(3) If f is continuous at the origin, F has a continuous extension $\tilde{F}$ to $\left(X,d\right)$ with $\tilde{F}\left(x\right)=0$ for every $x\in M$, and the metric space $\left(X,d\right)$ is complete, then $\left(X\setminus M,j_{F,f}\right)$ is also complete.

Proof. (1) Let $x\in X\setminus M$. Assume that $\underset{n\to \infty }{lim}{j}_{F,f}\left(z_n,x\right)=0$, where $z_n\in X\setminus M$ for every $n\ge 1$. The first inequality in Lemma 10 shows that $log\left(1+{\displaystyle \frac{f(d\left(\left(z_n,x\right)\right)}{F\left(x\right)}}\right)\le$$j_{F,f}\left(z_n,x\right)$ for every $n\ge 1$, therefore $\underset{n\to \infty }{lim}f\left(d\left(z_n,x\right)\right)=0$. By Lemma 1 (a), we have $\underset{n\to \infty }{lim}d\left(z_n,x\right)=0.$ As $x\in X\setminus M$ is arbitrary, it follows that the inverse of the map g is continuous, hence g is open.

(2)Let $x\in X\setminus M$. Assume that $\underset{n\to \infty }{lim}d\left(x_n,x\right)=0$, where $x_n\in X\setminus M$ for every $n\ge 1$. As $\underset{t\searrow 0}{lim}f\left(t\right)=0$ and $\left|F\left(x_n\right)-F\left(x\right)\right|\le f\left(d\left(x_n,x\right)\right)$ for all n, we get $\underset{n\to \infty }{lim}f\left(d\left(x_n,x\right)\right)=0$ and $\underset{n\to \infty }{lim}F\left(x_n\right)=F\left(x\right)>0$. Then $\underset{n\to \infty }{lim}{j}_{F,f}\left(x_n,x\right)=\underset{n\to \infty }{lim}log\left(1+{\displaystyle \frac{f\left(d(x_n,x)\right)}{min\left\{F\left(x_n\right),F\left(x\right)\right\}}}\right)=0$. As $x\in X\setminus M$ is arbitrary, it follows that g is continuous, hence g is a homeomorphism.

We estimate the linear dilatation of $g=1_{X\setminus M}$ at $x\in X\setminus M$. For all $r>0$, we have by definition

$$H_g\left(x,r\right)={\displaystyle \frac{sup\left\{j_{F,f}(x,y):d\left(x,y\right)\le r,y\in X\setminus M\right\}}{inf\left\{j_{F,f}(x,y):d\left(x,y\right)\ge r,y\in X\setminus M\right\}}}.$$

Using again the inequality $j_{F,f}\left(x,y\right)\ge log\left(1+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{F\left(x\right)}}\right)$, we obtain

$$inf\left\{j_{F,f}(x,y):d\left(x,y\right)\ge r,y\in X\setminus M\right\}\ge log\left(1+{\displaystyle \frac{f\left(r\right)}{F\left(x\right)}}\right).$$

From $max\left\{F\left(x\right),F\left(y\right)\right\}-min\left\{F\left(x\right),F\left(y\right)\right\}\le f\left(d(x,y)\right)$, it follows that $min\left\{F\left(x\right),F\left(y\right)\right\}\ge F\left(x\right)-f\left(d(x,y)\right)$, and if $f\left(d\right(x,y\left)\right)<F\left(x\right)$ this implies $j_{F,f}(x,y)\le log{\displaystyle \frac{F\left(x\right)}{F\left(x\right)-f\left(d\right(x,y\left)\right)}}$. Consequently, for every $r>0$ such that $f\left(r\right)<F\left(x\right)$, we have

$$sup\left\{j_{F,f}(x,y):d\left(x,y\right)\le r,y\in X\setminus M\right\}\le log{\displaystyle \frac{F\left(x\right)}{F\left(x\right)-f\left(r\right)}}.$$

So, for every $r>0$ such that $f\left(r\right)<F\left(x\right)$ we obtain

$$H_g\left(x,r\right)\le {\displaystyle \frac{log\frac{F\left(x\right)}{F\left(x\right)-f\left(r\right)}}{log\left(1+\frac{f\left(r\right)}{F\left(x\right)}\right)}}.$$

But $\underset{r\searrow 0}{lim}f\left(r\right)=0$, therefore

$$\underset{r\to 0}{limsup}{H}_g\left(x,r\right)\le \underset{r\to 0}{lim}{\displaystyle \frac{log\frac{F\left(x\right)}{F\left(x\right)-f\left(r\right)}}{log\left(1+\frac{f\left(r\right)}{F\left(x\right)}\right)}}=\underset{r\to 0}{lim}{\displaystyle \frac{\frac{F\left(x\right)}{F\left(x\right)-f\left(r\right)}-1}{\frac{f\left(r\right)}{F\left(x\right)}}}=1.$$

(3) Assuming that F has a continuous extension $\tilde{F}$ to X with $\tilde{F}\left(x\right)=0$ for every $x\in M$ and that $\left(X,d\right)$ is complete, we prove that $\left(X\setminus M,j_{F,f}\right)$ is complete.

Let ${\left(x_n\right)}_{n\ge 1}$ be a Cauchy sequence in $\left(X\setminus M,j_{F,f}\right)$. As $j_{F,f}\left(x,y\right)\ge \left|log{\displaystyle \frac{F\left(x\right)}{F\left(y\right)}}\right|$ for all $x,y\in X\setminus M$, ${\left(logF\left(x_n\right)\right)}_{n\ge 1}$ is a Cauchy sequence in $\mathbb{R}$. Then ${\left(logF\left(x_n\right)\right)}_{n\ge 1}$ is bounded: $\alpha =\underset{n\ge 1}{inf}$$logF\left(x_n\right)\in \mathbb{R}$ and $\beta =\underset{n\ge 1}{sup}$$logF\left(x_n\right)\in \mathbb{R}$. Then $e^{\alpha }=\underset{n\ge 1}{inf}$$F\left(x_n\right)\le \underset{n\ge 1}{sup}F\left(x_n\right)=e^{\beta }$. The first inequality in Lemma 10 implies

$$f\left(d\left(x_m,x_n\right)\right)\le F\left(x_m\right)\left(e^{j_{F,f}(x_m,x_n)}-1\right)\le e^{\beta }\left(e^{j_{F,f}(x_m,x_n)}-1\right)$$

(25)

for all $m,n\ge 1$. It follows that ${\left(x_n\right)}_{n\ge 1}$ is a Cauchy sequence in $\left(X,d\right)$. Indeed, if ${\left(x_n\right)}_{n\ge 1}$ is not a Cauchy sequence in $\left(X,d\right)$, then there exist ${\epsilon }_0>0$ and some sequences of integers, tending to infinity, ${\left(m_k\right)}_{k\ge 1}$ and ${\left(n_k\right)}_{k\ge 1}$, such that $d\left(x_{m_k},x_{n_k}\right)\ge {\epsilon }_0$ for all $k\ge 1$. As f is nondecreasing, for all $k\ge 1$ we have $f\left(d\left(x_{m_k},x_{n_k}\right)\right)\ge f\left({\epsilon }_0\right)>0$, hence,

$$j_{F,f}\left(x_{m_k},x_{n_k}\right)\ge log(1+e^{-\beta }f\left({\epsilon }_0\right)).$$

We used inequality (25). The latter inequality contradicts the assumption that ${\left(x_n\right)}_{n\ge 1}$ is a Cauchy sequence in $\left(X\setminus M,j_{F,f}\right)$.

By our assumption, the metric space $\left(X,d\right)$ is complete. Let $x\in X$ be such that $\underset{n\to \infty }{lim}d\left(x_n,x\right)=0$. As f is continuous at the origin, $\underset{n\to \infty }{lim}f\left(d\left(x_n,x\right)\right)=0.$

Since $\tilde{F}$ is continuous on X, $\tilde{F}\left(x\right)=\underset{n\to \infty }{lim}\tilde{F}\left(x_n\right)=\underset{n\to \infty }{lim}F\left(x_n\right)\ge e^{\alpha }>0$, hence $x\in X\setminus M$ and $\tilde{F}\left(x\right)=F\left(x\right)$. Finally, $\underset{n\to \infty }{lim}{j}_{F,f}\left(x_n,x\right)=\underset{n\to \infty }{lim}log\left(1+{\displaystyle \frac{f\left(d(x_n,x)\right)}{min\left\{F\left(x_n\right),F\left(x\right)\right\}}}\right)=0$ , hence ${\left(x_n\right)}_{n\ge 1}$ converges to x in $\left(X\setminus M,j_{F,f}\right)$. □

We provide some natural assumptions on $F:X\setminus M\to (0,\infty )$ and $f:[0,\infty )\to [0,\infty )$ to ensure the Gromov hyperbolicity of modified Vuorinen’s metric $j_{F,f}$ on $X\setminus M$.

Theorem 7.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive, and there exists $\lambda \ge 1$ such that $f\left(t\right)\le \lambda t$ for all $t\in [0,\infty )$. Let $F:X\setminus M\to (0,\infty )$ and $G\subset X\setminus M$ such that, for all $x,y\in G$,

$$\left|F\left(x\right)-F\left(y\right)\right|\le f\left(d(x,y)\right)\mathrm{and}d(x,y)\le 2max\left\{F\left(x\right),F\left(y\right)\right\}.$$

(26)

Then the metric $j_{F,f}$ is Gromov hyperbolic on G, with a constant $\delta \le log\left(1+2\lambda \right)$.

Proof. 

By Theorem 5, $j_{F,f}$ is a metric on G, as the first inequality in (26) holds.

By Lemma 10, $j_{F,f}(x,y)\ge \left|log{\displaystyle \frac{F\left(x\right)}{F\left(y\right)}}\right|=log{\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}$.

For all $x,y\in G$, from our assumptions on F and f it follows that

${\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right))\right\}}}\le 2\lambda {\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}$. But $1\le {\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}$, therefore, adding the latter two inequalities and taking logarithms we obtain

$$j_{F,f}(x,y)\le log{\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}+log(1+2\lambda ).$$

Therefore, for all $x,y\in G$,

$$log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}\right)\le j_{F,f}(x,y)\le log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}\right)+log\left(1+2\lambda \right).$$

As the Barbilian’s logarithmic oscillation $log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}\right)$ is $0-$hyperbolic in the sense of Gromov on $X\setminus M$, therefore on G, taking $\mathrm{\Phi }=1_G$ and $\alpha =0$, $\beta =log\left(1+2\lambda \right),\gamma =0$, $\delta =0$ in Lemma 8 it follows that the metric $j_{F,f}$ is Gromov hyperbolic on G, with constant $\delta \le log\left(1+2\lambda \right)$. □

The following consequence of Theorem 7 generalizes [14], Theorem 3, Sufficiency and [6], Theorem 6.3 (b).

Corollary 3.  

Let $\left(X,d\right)$ be a metric space. Given $p\in X$, denote $\delta \left(x\right)=d(x,p),x\in X$. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive, and there exists $\lambda \ge 1$ such that $t\le f\left(t\right)\le \lambda t$ for all $t\in [0,\infty )$. For $x,y\in X\setminus \left\{p\right\}$ we define

$$j_{X\setminus \left\{p\right\},f}(x,y)=log\left(1+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{\delta \left(x\right),\delta \left(x\right)\right\}}}\right).$$

Then $j_{X\setminus \left\{p\right\},f}$ is a $log\left(1+2\lambda \right)-$hyperbolic metric on $X\setminus \left\{p\right\}$. In particular, if f is the identity of $[0,\infty )$, then the above metric is $log3-$hyperbolic.

Proof. 

Here $M=\left\{p\right\}$ and $F\left(x\right)=d(x,p)$. Let $x,y\in X\setminus \left\{p\right\}$. By triangle inequality, $d(x,y)\le F\left(x\right)+F\left(y\right)\le 2max\left\{F\left(x\right),F\left(y\right)\right\}$, therefore the second inequality in (26) holds. Again by triangle inequality, $\left|F\left(x\right)-F\left(y\right)\right|\le d(x,y)$, but $d(x,y)\le f\left(d\right(x,y\left)\right)$, therefore the first inequality in (26) is also true. The claim follows using Theorem 7. □

6. Vuorinen’s Stabilizing Metric

In the following, we extend the Vuorinen’s stabilizing metric introduced by Boskoff and Suceavă [6], page 727, defined for $x,y\in {\mathbb{R}}^n\setminus M$ by

$$d_{VH,s}(x,y)=log\left({\displaystyle \frac{max\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}{min\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}}+{\displaystyle \frac{d(x,y)}{min\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}}\right),$$

where $M\subset {\mathbb{R}}^n$ is a closed set and ${\delta }_M\left(z\right)=\mathrm{dist}\left(z,M\right)=inf\left\{\left|z-m\right|:m\in M\right\}$ for $z\in X$.

In [6], Theorem 6.1 it was proved in the special case $M=\left\{0_{{\mathbb{R}}^n}\right\}$ that $d_{VH,s}$ is a metric on ${\mathbb{R}}^n\setminus M$, which is Gromov hyperbolic with a constant $\delta \le log3$.

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Let $F:\left(X\setminus M,d\right)\to (0,\infty )$ and $f:[0,\infty )\to [0,\infty )$. For $x,y\in X\setminus M$ we define

$$j_{F,f,s}(x,y)=log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\right).$$

Obviously, $j_{F,f,s}(x,y)\ge j_{F,f}(x,y)$ for all $x,y\in X\setminus M$.

Theorem 8.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive. Let $F:X\setminus M\to (0,\infty )$ such that, for all $x,y\in X\setminus M$,

$$\left|F\left(x\right)-F\left(y\right)\right|\le f\left(d(x,y)\right).$$

(27)

Then $j_{F,f,s}$ is a metric on $X\setminus M$.

Proof. 

Obviously, $j_{f,s}$ satisfies the axioms of nonnegativity, separation and symmetry. The triangle inequality $j_{F,f,s}(x,y)\le j_{F,f,s}(x,z)+j_{F,f,s}(z,y)$ for any $x,y,z\in X\setminus M$ is equivalent to

$$\begin{array}{ccc}\hfill {\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}& \le & \left({\displaystyle \frac{max\left\{F\left(x\right),F\left(z\right)\right\}}{min\left\{F\left(x\right),F\left(z\right)\right\}}}+{\displaystyle \frac{f\left(d\right(x,z\left)\right)}{min\left\{F\left(x\right),F\left(z\right)\right\}}}\right)\hfill \\ & & \times \left({\displaystyle \frac{max\left\{F\left(z\right),F\left(y\right)\right\}}{min\left\{F\left(z\right),F\left(y\right)\right\}}}+{\displaystyle \frac{f\left(d\right(z,y\left)\right)}{min\left\{F\left(z\right),F\left(y\right)\right\}}}\right).\hfill \end{array}$$

(28)

Using inequality (24) and the inequalities ${\displaystyle \frac{max\left\{F\left(x\right),F\left(z\right)\right\}}{min\left\{F\left(x\right),F\left(z\right)\right\}}}\ge 1$, ${\displaystyle \frac{max\left\{F\left(z\right),F\left(y\right)\right\}}{min\left\{F\left(z\right),F\left(y\right)\right\}}}\ge 1$ we get

$$\begin{array}{ccc}\hfill {\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}& \le & {\displaystyle \frac{max\left\{F\left(z\right),F\left(y\right)\right\}}{min\left\{F\left(z\right),F\left(y\right)\right\}}}{\displaystyle \frac{f\left(d\right(x,z\left)\right)}{min\left\{F\left(x\right),F\left(z\right)\right\}}}+{\displaystyle \frac{max\left\{F\left(x\right),F\left(z\right)\right\}}{min\left\{F\left(x\right),F\left(z\right)\right\}}}{\displaystyle \frac{f\left(d\right(z,y\left)\right)}{min\left\{F\left(z\right),F\left(y\right)\right\}}}\hfill \\ & & +{\displaystyle \frac{f\left(d\right(x,z\left)\right)}{min\left\{F\left(x\right),F\left(z\right)\right\}}}{\displaystyle \frac{f\left(d\right(z,y\left)\right)}{min\left\{F\left(z\right),F\left(y\right)\right\}}}.\hfill \end{array}$$

Recall from (7) that for every function $F:X\setminus M\to (0,\infty )$ and all $x,y,z\in X\setminus M$ we have

$${\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\le {\displaystyle \frac{max\left\{F\left(x\right),F\left(z\right)\right\}}{min\left\{F\left(x\right),F\left(z\right)\right\}}}{\displaystyle \frac{max\left\{F\left(z\right),F\left(y\right)\right\}}{min\left\{F\left(z\right),F\left(y\right)\right\}}}.$$

Adding the latter two inequalities we obtain the claim (28). □

Example 1.  

If $F\left(x\right)={\delta }_M\left(x\right)=\mathrm{dist}(x,M)$ for all $x\in X\setminus M$, and the function f from Theorem 8 satisfies $f\left(t\right)\ge t$ for all $t\in [0,\infty )$, then inequality (27) holds for all $x,y\in X\setminus M$, hence $j_{F,f,s}$ is a metric on $X\setminus M$. For $X={\mathbb{R}}^n$ we recover Theorem 6.1 (a) in [6].

We prove that Vuorinen’s stabilizing metric $j_{F,f,s}$ and Vuorinen’s modified metric $j_{F,f}$ on $X\setminus M$ are bi-Lipschitz equivalent, under the assumption (27).

Theorem 9.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty proper subset of X. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive. Let $F:X\setminus M\to (0,\infty )$ such that, for all $x,y\in X\setminus M$,

$$\left|F\left(x\right)-F\left(y\right)\right|\le f\left(d(x,y)\right).$$

Then $j_{F,f}(x,y)\le j_{F,f,s}(x,y)\le 2j_{F,f}(x,y)$ for all $x,y\in X\setminus M$.

Proof. 

Let $x,y\in X\setminus M$. We noticed that $j_{F,f}(x,y)\le j_{F,f,s}(x,y)$, without any assumptions on $F:X\setminus M\to (0,\infty )$ and $f:[0,\infty )\to [0,\infty )$.

Using the identity ${\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right)\right\}}}+{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}=1+{\displaystyle \frac{\left|F\left(x\right)-F\left(y\right)\right|+f\left(d(x,y)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}$ and the assumption (27) $\left|F\left(x\right)-F\left(y\right)\right|\le f\left(d(x,y)\right)$, we obtain $j_{F,f,s}(x,y)\le log\left(1+2{\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right)\right\}}}\right)$. But $log(1+2t)\le 2log(1+t)$ for all $t>-{\displaystyle \frac{1}{2}}$. The latter two inequalities imply $j_{F,f,s}(x,y)\le 2j_{F,f}(x,y)$. □

Corollary 4.  

Let $\left(X,d\right)$, $M\subset X$, $f:[0,\infty )\to [0,\infty )$ and $F:X\setminus M\to (0,\infty )$ satisfying the assumptions from Theorem 9. In addition, assume that f is continuous at the origin. Consider the identity map $u:=1_{X\setminus M}:\left(X\setminus M,d\right)\to \left(X\setminus M,j_{F,f,s}\right)$.

(1) u is 2-quasiconformal;

(2) If F has a continuous extension $\tilde{F}$ to $\left(X,d\right)$ with $\tilde{F}\left(x\right)=0$ for every $x\in M$, and the metric space $\left(X,d\right)$ is complete, then $\left(X\setminus M,j_{F,f,s}\right)$ is also complete.

Proof. (1) By Theorem 9, the metrics $j_{F,f,s}$ and $j_{F,f}$ on $X\setminus M$ are bi-Lipschitz equivalent. Then the spaces $\left(X\setminus M,j_{F,f,s}\right)$ and $\left(X\setminus M,j_{F,f,s}\right)$ share the same Cauchy sequences and a sequence converges in one metric if and only if it converges in the other, and it converges to the same point.

Applying Theorem 6 (3), it follows that $\left(X\setminus M,j_{F,f,s}\right)$ is complete, hence $\left(X\setminus M,j_{F,f,s}\right)$ is also complete.

(2) Let $g:=1_{X\setminus M}:\left(X\setminus M,d\right)\to \left(X\setminus M,j_{F,f}\right)$. As g is a homeomorphism and the metrics $j_{F,f,s}$ and $j_{F,f}$ induce the same topology, u is also a homeomorphism.

From the inequalities $j_{F,f}\le j_{F,f,s}\le 2j_{F,f}$ on $X\setminus M$ we get, for all $x\in X\setminus M$ and $r>0$, the following estimates:

$inf\left\{j_{F,f},s(x,y):d\left(x,y\right)\ge r,y\in X\setminus M\right\}\ge inf\left\{j_{F,f}(x,y):d\left(x,y\right)\ge r,y\in X\setminus M\right\}$ and $sup\left\{j_{F,f},s(x,y):d\left(x,y\right)\le r,y\in X\setminus M\right\}\le 2sup\left\{j_{F,f}(x,y):d\left(x,y\right)\le r,y\in X\setminus M\right\}$.

Then $H_u(x,r)\le 2H_g(x,r)$ for all $x\in X\setminus M$ and $r>0$.

By Theorem 6 (2), $\underset{r\to 0}{limsup}{H}_g(x,r)\le 1$. It follows that $\underset{r\to 0}{limsup}{H}_u(x,r)\le 2$. □

We provide some natural assumptions on $F:X\setminus M\to (0,\infty )$ and $f:[0,\infty )\to [0,\infty )$ in order to ensure the Gromov hyperbolicity of Vuorinen’s stabilizing metric $j_{F,f,s}$ on $X\setminus M$.

Theorem 10.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty closed proper subset of X. Assume that $f:[0,\infty )\to [0,\infty )$ vanishes only at the origin, is nondecreasing and subadditive, and there exists $\lambda >0$ such that $f\left(t\right)\le \lambda t$ for all $t\in [0,\infty )$. Let $F:X\setminus M\to (0,\infty )$ and $G\subset X\setminus M$ such that, for all $x,y\in G$,

$$\left|F\left(x\right)-F\left(y\right)\right|\le f\left(d(x,y)\right)\mathrm{and}d(x,y)\le 2max\left\{F\left(x\right),F\left(y\right)\right\}.$$

(29)

Then the metric $j_{F,f,s}$ is Gromov hyperbolic on G, with a constant $\delta \le log\left(1+2\lambda \right)$.

Proof. 

By Theorem 8, $j_{F,f,s}$ is a metric on $X\setminus M$, hence on G.

For all $x,y\in G$ it follows from our assumptions on F and f that

$d(x,y)\le 2max\left\{F\left(x\right),F\left(y\right)\right\}$, hence, ${\displaystyle \frac{f\left(d\right(x,y\left)\right)}{min\left\{F\left(x\right),F\left(y\right))\right\}}}\le 2\lambda {\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}.$

Therefore, for all $x,y\in G$,

$$log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}\right)\le j_{F,f,s}(x,y)\le log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}\right)+log\left(1+2\lambda \right).$$

Since the Barbilian logarithmic oscillation $log\left({\displaystyle \frac{max\left\{F\left(x\right),F\left(y\right)\right\}}{min\left\{F\left(x\right),F\left(y\right))\right\}}}\right)$ is $0-$hyperbolic in the sense of Gromov on $X\setminus M$, therefore on G, taking $\mathrm{\Phi }=1_G$ and $\alpha =0$, $\beta =log\left(1+2\lambda \right),\gamma =0$, $\delta =0$ in Lemma 8 it follows that the $j_{F,f,s}$ is Gromov hyperbolic on G, with a constant $\delta \le log\left(1+2\lambda \right)$. □

Corollary 5.  

Let $\left(X,d\right)$ be a metric space and M be a nonempty closed proper subset of X. Denote ${\delta }_M\left(x\right)=\mathrm{dist}(x,M)$, $x\in X$. For $x,y\in X\setminus M$ we define

$$j_{M,s}(x,y)=log\left({\displaystyle \frac{max\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}{min\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}}+{\displaystyle \frac{d(x,y)}{min\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}}}\right).$$

Let $G\subset X\setminus M$. If $d(x,y)\le 2max\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}$ for all $x,y\in G$, then $j_{M,s}$ is a $log3-$hyperbolic metric on G.

Proof. 

In Theorem 8 and Theorem 10 let $F={\delta }_M$ and f be the identity of $[0,\infty )$, hence, $\lambda =1$. Since $d_M$ is $1-$Lipschitz on X, $j_{M,s}$ is a metric on $X\setminus M$ by Theorem 8 and the first inequality in (29) holds for all $x,y\in X$. The assumption $d(x,y)\le {\delta }_M\left(x\right)+{\delta }_M\left(y\right)$ for all $x,y\in G$ shows that the second inequality in (29) holds for all $x,y\in G$. The claim follows, since $log\left(1+2\lambda \right)=log3.$ □

Example 2.  

If $M=\left\{p\right\}\subset X$, then

$$d(x,y)\le d(x,p)+d(p,y)\le 2max\left\{d(x,p),d(p,y)\right\}=2max\left\{{\delta }_M\left(x\right),{\delta }_M\left(y\right)\right\}$$

holds for all $x,y\in X\setminus \left\{p\right\}$, hence $j_{M,s}$ is a $log3-$hyperbolic metric on $X\setminus \left\{p\right\}$. For $X={\mathbb{R}}^n$ and $p=0_{{\mathbb{R}}^n}$ we recover Theorem 6.1 (b) in [6].

Remark 4.  

The assumption that $d(x,y)\le 2max\left\{F\left(x\right),F\left(y\right)\right\}$ for all $x,y\in G$, used in Theorem 7 and 10 is restrictive in the case $F\left(z\right)={\delta }_{\partial G}\left(z\right)=\mathrm{dist}(z,\partial G)$, $z\in X$, since it implies that G has a single boundary point. Indeed, for any $\epsilon >0$ we find $a_{\epsilon },b_{\epsilon }\in \partial G$ such that $d(a_{\epsilon },b_{\epsilon })>\mathrm{diam}(\partial G)-\epsilon$, as well as $x_{\epsilon },y_{\epsilon }\in G$ such that $d(x_{\epsilon },a_{\epsilon })<\epsilon$ and $d(y_{\epsilon },b_{\epsilon })<\epsilon$. By triangle inequality and the symmetry of the distance, $d(x_{\epsilon },y_{\epsilon })\ge d(a_{\epsilon },b_{\epsilon })-d(x_{\epsilon },a_{\epsilon })-d(y_{\epsilon },b_{\epsilon })$, hence $d(x_{\epsilon },y_{\epsilon })\ge \mathrm{diam}(\partial G)-3\epsilon$. On the other hand, if we assume that $d(x,y)\le 2max\left\{{\delta }_{\partial G}\left(x\right),{\delta }_{\partial G}\left(y\right)\right\}$ for all $x,y\in G$, we get $d(x_{\epsilon },y_{\epsilon })\le 2max\left\{d(x_{\epsilon },a_{\epsilon }),d(y_{\epsilon },b_{\epsilon })\right\}<2\epsilon$. We conclude that $\mathrm{diam}(\partial G)-3\epsilon <2\epsilon$ for every $\epsilon >0$, therefore $\mathrm{diam}(\partial G)=0$.

Recall that in the case where $X={\mathbb{R}}^n$ with the Euclidean metric, $G{\mathbb{R}}^n$ is open, $F={\delta }_{\partial G}$ and f is the identity of $[0,\infty )$, Hästö proved in [14], Theorem 3 that $j_{F,f}$ is Gromov hyperbolic on G if and only if G has a single boundary point. Similarly, Boskoff and Suceavă proved in [6], Theorem 6.2, in the case where $X={\mathbb{R}}^n$ with the Euclidean metric, $M{\mathbb{R}}^n$ is closed, $F={\delta }_M$ and f is the identity of $[0,\infty )$, that Vuorinen’ stabilizing metric $j_{M,s}$ is Gromov hyperbolic on $X\setminus M$ only if M has exactly one point.