In this paper, using fixed point methods we investigate Lie $*$--homomorphisms between Lie C*--algebras, and Lie $*$--derivations on Lie C*--algebras associated with the generalized Jensen--type functional equation $\mu f(\frac{\sum^{n}_{i=1}x_{i}}{n})+\mu \sum^{n}_{j=2}f(\frac{\sum^{n}_{i=1,i\neq j}x_{i}-(n-1)x_{j}}{n})-f(\mu x_1)=0 .$
