  We explicitly find the minima as well as the minimum points of the geodesic
length functions for the family of filling (hence non-simple) closed curves,
$a^2b^n$ ($n\ge 3$), on a complete one-holed hyperbolic torus in its relative
Teichm\"uller space, where $a, b$ are simple closed curves on the one-holed
torus which intersect exactly once transversely. This provides concrete
examples for the problem to minimize the geodesic length of a fixed filling
closed curve on a complete hyperbolic surface of finite type in its relative
Teichm\"uller space.
