The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed in [T.-P. Liu, T. Yang, A new entropy functional for scalar conservation laws, Comm. Pure Appl. Math. 52 (1999) 14271442] for the L 1 stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, we introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [J.-L. Hua, Z.-H. Jiang, T. Yang, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, preprint] for general scalar conservation laws. This functional improves the one given in [H.-X. Liu, T. Yang, A