I'm reading a paper on Competition in Two-Sided Markets. The model is a Hotelling-type model, with consumers on some interval, choosing their preferred firm based on price and 'distance' to the different firms (see below). In it, there's an assumption made that guarantees that there's no single monopoly that serves all consumers, which amounts to the condition:
But I'm not sure how this is derived. Annoyingly, the paper says: "it turns out that the necessary and sufficient condition for a market sharing equilibrium is $4t_1t_2>(\alpha_1\alpha_2)^2$", without providing the derivation. I'd appreciate a hint on how to derive this, or an actual derivation of this condition.
These screenshots are from page 8 from Competition in Two-Sided Markets.