Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma\_i; i = 1, 2, 3$ are the geodesic boundaries of length $l\_i; i=1, 2, 3$ respectively. Now consider a essential simple arc $\sigma$ in $Y$ with end points on a same boundary component of $Y$ and $l$ denote the length of unique geodesic in the homotopy class of $\sigma$. Then, my question is weather the inequality, $l$ $\geq$ $\frac{1}{2}$ min{$l\_i | i= 1, 2, 3$} hold or not for every pair of pants $Y$. 



Hyperbolic pair of pants.
If you keep the lengths of two cuffs fixed (say equal to L) and let the third one go to infinity (say equal to R), then the length $l$ of the ortho-geodesic with endpoints on the third cuff goes to zero. There is a trigonometric formula to see this. Cut the pants into two right-angled hexagons then further into four pentagons with the ortho-geodesic. Formula 2.6.17 for right-angled pentagons in Chapter 2 of Thurston's notes gives $\sinh(R/4) \sinh (l/2) = \cosh(L/2)$. Therefore no such bound holds.


