  In this paper, we show that the extremal length functions on Teichm\"uller
space are log-plurisubharmonic. As a corollary, we obtain an alternative proof
of L.Liu and W.Su's results on the plurisubharmonicity of extremal length
functions. We also obtain alternative proofs of S.Krushkal's results that a
function defined by the Teichm\"uller distance from a reference point is
plurisubharmonic, and the Teichm\"uller space is hyperconvex. To show the
log-plurisubharmonicity, we give an explicit formula of the Levi form of the
extremal length functions in generic case.
