Metric transforms yielding Gromov hyperbolic spaces

A real valued function $\phi$ of one variable is called a metric transform if for every metric space $(X,d)$ the composition $d_{\phi }=\phi \circ d$ is also a metric on $X$. We give a complete characterization of the class of approximately nondecreasing, unbounded metric transforms $\phi$ such that the transformed Euclidean half line $([0,\mathrm{\infty }),\mathrm{\mid }\cdot {\mathrm{\mid }}_{\phi })$ is Gromov hyperbolic. As a consequence, we obtain metric transform rigidity for roughly geodesic Gromov hyperbolic spaces, that is, if $(X,d)$ is any metric space containing a rough geodesic ray and $\phi$ is an approximately nondecreasing, unbounded metric transform such that the transformed space $(X,d_{\phi })$ is Gromov hyperbolic and roughly geodesic then $\phi$ is an approximate dilation and the original space $(X,d)$ is Gromov hyperbolic and roughly geodesic.