Bounds of the Gromov product associated with quasi-geodesics

Question

Let $l_1$ be a quasi-geodesic with endpoints $x_0,x_1$ and $l_2$ be a quasi-geodesic with endpoints $x_1,x_2$ in a $\delta$-hyperbolic space. Let $y_i\in l_i$ be a point ($i=1,2$). Suppose $d(y_1,x_0),d(y_2,x_2)<H$, then is that possible to bound the Gromov product $(y_1,y_2)_{x_1}$ in terms of $(x_0,x_2)_{x_1}$ and $H$ (from above)?

Answer 1

You can directly calculate it from the Gromov product just using the triangle inequality:

$$(y_1,y_2)_{x_1} = \frac{1}{2}(d(y_1,x_1)+d(y_2,x_1)- d(y_1,y_2)) \leq \frac{1}{2}(d(x_0,x_1) + H + d(x_1,x_2) + H - d(y_1,y_2))$$

And from $d(x_0,x_2) \leq 2H + d(y_1,y_2)$ it then follows that: $$(y_1,y_2)_{x_1} \leq \frac{1}{2}(4H + d(x_0,x_1) + d(x_1,x_2) - d(y_1,y_2)) = (x_0,x_2)_{x_1} + 2H$$