Precise estimates of invariant distances on strongly pseudoconvex domains

Studying the behavior of real and complex geodesics we provide sharp estimates for the Kobayashi distance, the Lempert function, and the Carathéodory distance on ${\mathcal{C}}^{2,\alpha }$-smooth strongly pseudoconvex domains. Similar estimates are also provided for the Bergman distance on strongly pseudoconvex domains with ${\mathcal{C}}^{3,1}$-boundary.