  In this paper, we study orbital stability of peakons for the generalized
modified Camassa-Holm (gmCH) equation, which is a natural higher-order
generalization of the modified Camassa-Holm (mCH) equation, and admits
Hamiltonian form and single peakons. We first show that the single peakon is
the usual weak solution of the PDEs. Some sign invariant properties and
conserved densities are presented. Next, by constructing the corresponding
auxiliary function $h(t,\,x)$ and establishing a delicate polynomial inequality
relating to the two conserved densities with the maximal value of approximate
solutions, the orbital stability of single peakon of the gmCH equation is
verified. We introduce a new approach to prove the key inequality, which is
different from that used for the mCH equation. This extends the result on the
stability of peakons for the mCH equation (Comm. Math. Phys., 322:967-997,
2013) successfully to the higher-order case, and is helpful to understand how
higher-order nonlinearities affect the dispersion dynamics.
