Warped product spaces: Gromov hyperbolicity and identification of the visual boundary

In this paper, we consider warped product spaces $X\times_{\varphi}Y$, where $X$ is a complete geodesic Gromov hyperbolic space, $Y$ is a compact geodesic metric space, and the warping function $\varphi$ satisfies suitable exponential growth conditions. We prove that the warped product is Gromov hyperbolic and derive an explicit estimate for its hyperbolicity constant. We further establish a homeomorphism between its Gromov boundary and $\partial_GX\times Y$, with an explicit comparison formula for the visual metric on the Gromov boundary of $X\times_\varphi Y$ in terms of the visual metric $d_{\varepsilon, X}$ on $\partial_GX$ and $d_Y$ for suitable $\varepsilon>0$.