  \noindent We study the concept of the complementarity, introduced by Bagchi
and Quesne in [Phys. Lett. A {\bf 301}, 173 (2002)], between pseudo-Hermiticity
and weak pseudo-Hermiticity in a rigorous mathematical viewpoint of coordinate
transformations when a system has a position-dependent mass. We first
determine, under the modified-momentum, the generating functions identifying
the complexified potentials $V_\pm(x)$ under both concepts of
pseudo-Hermiticity $\widetilde\eta_+$ (resp. weak pseudo-Hermiticity
$\widetilde\eta_-$). We show that the concept of complementarity can be
understood and interpreted as a coordinate transformation through their
respective generating functions. As consequence, a similarity transformation
which implements coordinate transformations is obtained. We show that the
similarity transformation is set up as fundamental relationship connecting both
$\widetilde\eta_+$ and $\widetilde\eta_-$. A special factorization
$\eta_+=\eta_-^\dagger \eta_-$ is discussed in the case of a constant mass and
some B\"acklund transformations are derived.
