Busemann functions and inequalities

Question

Let $D$ is the hyperbolic unit disk. Let $\alpha,\,\beta\in S^1$, where $S^1$ is the boundary of $D$. Let $w\in D$. I know that Busemann function for hyperbolic disk is $$B(w,\alpha)=\ln\frac{1-|w|^2}{|\alpha-w|^2}.$$ I know that there is the following inequality, $$|B(w,\alpha)-B(v,\alpha)|\leq L(w,v),$$ where $L(w,v)$ is the geodesic length between points $w$ and $v$. Is it possible to write down something similar for different bounday points, i.e. $|B(w,\alpha)-B(w,\beta)|\vee L(\alpha,\beta)$, where $\vee$ is comparison sign?