Comparing hyperbolic distance with Kra's distance on the unit disk

Abstract

In this paper, Kra's distance $d_{K}$ and the hyperbolic distance $d_{\mathbb{D}}$ are compared on the unit disk $\mathbb{D}$. It is shown that $2d_{K} < d_{\mathbb{D}} < (\pi^{2}/8)\exp{d_{K}}$ on $\mathbb{D} \times \mathbb{D} \setminus \{\text{diagonal}\}$, where the constants $2$ and $\pi^{2}/8$ are sharp. As a consequence, this result gives a negative answer to a question posed by Martin [7] in a stronger sense.