The author proves the following theorems. Theorem 1. Let (M,g) and (N,h) be complete Riemannian manifolds with \(Ric^ M\geq -B^ 2\) and \(Riem^ N\leq A^ 2\) where \(A>0\) and \(B>0\). Suppose that \(f: M\to N\) is a bounded harmonic map such that \(f(M)\subseteq B_{\rho_ 0}(Q)\), where \(\rho_ 0<\frac{\pi}{2A}\), and the intersection of \(B_{\rho_ 0}(Q)\) and the cut locus of Q is empty. Then \(f^*ds^ 2_ N/\cos (A\rho)\leq 2KB^ 2/A^ 2\cos (A\rho_ 0)ds^ 2_ M\), where \(\rho\) \((\tilde Q)=dist_ N(\tilde Q,Q)\) and \(K=\min \{m,n\}\). Theorem 2. Let (M,g) and (N,h) be complete Riemannian manifolds with \(Ric^ M\geq -B^ 2\) where \(B>0\), and \(Riem^ N\leq 0\). Suppose that \(f: M\to N\) is a bounded harmonic map with \(f(M)\subseteq B_{\rho_ 0}(Q)\) where \(\rho_ 0<\infty\) and the intersection of \(B_{\rho_ 0}(Q)\) and the cut locus of Q is empty. Then \(f^*ds^ 2_ N\leq (\rho^ 2_ 0-\rho^ 2)B^ 2ds^ 2_ M\). Theorem 1 and 2 imply the following Liouville Theorem. Theorem 3. Let (M,g) and (N,h) be complete Riemannian manifolds with \(Ric^ M\geq 0\) and \(Riem^ N\leq A^ 2\) where \(A>0\). Suppose that \(f: M\to N\) is a bounded harmonic map. Then f is constant.