Complementarity vs coordinate transformations: Mapping between pseudo-Hermiticity and weak pseudo-Hermiticity

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