  In this paper, we introduce a new energy density function $\mathscr Y$ on the
projective bundle $\mathbb{P}(T_M)\>M$ for a smooth map $f:(M,h)\>(N,g)$
between Riemannian manifolds $$\mathscr Y=g_{ij}f^i_\alpha f^j_\beta
\frac{W^\alpha W^\beta}{\sum h_{\gamma\delta} W^\gamma W^\delta}.$$ We get new
Hessian estimates to this energy density and obtain various new Liouville type
theorems for holomorphic maps, harmonic maps and pluri-harmonic maps. For
instance, we show that there is no non-constant holomorphic map from a compact
\emph{Hermitian manifold} with positive (resp. non-negative) holomorphic
sectional curvature to a \emph{Hermitian manifold} with non-positive (resp.
negative) holomorphic sectional curvature.
