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In this paper, we prove that the class of bi-f-harmonic maps and that off-biharmonic maps from a conformal manifold of dimension not equal to 2 are thesame (Theorem 1.1). We also give several results on nonexistence of properbi-f-harmonic maps and f-biharmonic maps from complete Riemannian manifoldsinto nonpositively curved Riemannian manifolds. These include: anybi-f-harmonic map from a compact manifold into a non-positively curved manifoldis f-harmonic (Theorem 1.6), and any f-biharmonic (respectively, bi-f-harmonic)map with bounded f and bounded f-bienrgy (respectively, bi-f-energy) from acomplete Riemannian manifold into a manifold of strictly negative curvature hasrank < 2 everywhere (Theorems 2.2 and 2.3).

Some remarks on bi-f-harmonic maps and f-biharmonic maps