Kobayashi-Royden vs. Hahn pseudometric in ℂ²

For a domain D ⊂ ℂ the Kobayashi-Royden ϰ and Hahn h pseudometrics are equal iff D is simply connected. Overholt showed that for ${\displaystyle D\subset {ℂ}^n}$, n ≥ 3, we have ${\displaystyle h_D\equiv {\varkappa }_D}$. Let D₁, D₂ ⊂ ℂ. The aim of this paper is to show that ${\displaystyle h_{D₁\times D₂}}$ iff at least one of D₁, D₂ is simply connected or biholomorphic to ℂ \ {0}. In particular, there are domains D ⊂ ℂ² for which ${\displaystyle h_D\not\equiv {\varkappa }_D}$.