  The observed charged lepton masses satisfy the relations $K \equiv (m_e
+m_\mu+m_\tau)/(\sqrt{m_e} +\sqrt{m_\mu} +\sqrt{m_\tau})^2 =2/3$ and $\kappa
\equiv \sqrt{m_e m_\mu m_\tau}/(\sqrt{m_e} +\sqrt{m_\mu} +\sqrt{m_\tau})^3
=1/486$ with great accuracy. These parameters are given as $K=( {\rm Tr}[\Phi
\Phi])/
  ({\rm Tr}[\Phi ])^2$ and $\kappa = {\rm det} \Phi/({\rm Tr}[\Phi ])^3$ if the
charged lepton masses $m_{ei}$ are given by $m_{ei} \propto \sum_k \Phi_i^{\ k}
\Phi_k^{\ i}$ where $\Phi$ is a U(3)-family nonet scalar. Simple scalar
potential forms to realize the relations have been already proposed in
non-supersymmetric scenarios, but the potential forms are not stable against
the renormalization group effects. In this paper, we examine supersymmetric
scenarios and find that the parameters $K$ and $\kappa$ are made stable against
the effects in a very nontrivial way, even though the superpotential itself (in
the canonical basis) suffers the usual corrections. We also show possible
simple superpotential forms for the relations.
