The kernel function determines the filtering robustness, as the convex function can not redescend, which degrades the robustness of Huber-based filters, whereas the non-convex kernel functions with the redescending ability can improve filtering robustness, yet they tend to induce the estimation to fall into a local minimum. This work investigates convex and non-convex kernel functions from robustness and stability perspectives, respectively. To improve the ability of robust filters to the high level of non-Gaussian observation noise, a mixed convex and non-convex robust function strategy is presented. To avoid the matrix singularity problem by applying the mixed strategy, which is induced by the non-convex function, the fixed-point iteration-based generalized M-estimation is transformed into an information filtering form. The simulation results show that, under different levels of heavy-tailed non-Gaussian noise, the mixed strategy can avoid the local minimum by applying a single non-convex function, and further improve the convex-function-based filtering accuracy. Therefore, the mixed strategy can comprehensively improve the efficiency of the fixed-point iteration-based generalized M-estimation.