  Both bi-harmonic map and $f$-harmonic map have nice physical motivation and
applications. In this paper, by combination of these two harmonic maps, we
introduce and study $f$-bi-harmonic maps as the critical points of the
$f$-bi-energy functional $\frac{1}{2}\int_M f|\tau (\phi)|^2dv_{g}$. This class
of maps generalizes both concepts of harmonic maps and bi-harmonic maps. We
first derive the $f$-biharmonic map equation and then use it to study
$f$-bi-harmonicity of some special maps, including conformal maps between
manifolds of same dimensions, some product maps between direct product manifold
and singly warped product manifold, some projection maps from and some
inclusion maps into a warped product manifold.
