  In this paper, we investigate the orbital stability problem of peakons for a
modified Camassa-Holm equation with both quadratic and cubic nonlinearity. This
equation was derived from integrable theory and admits peaked soliton (peakon)
and multipeakon solutions. By constructing two suitable piecewise functions, we
establish the polynomial inequality relating to two conserved quantities and
the maximum of the solution to this equation. The error estimate between the
maximum of the solution and the peakon then follows from the structure of the
polynomial inequality. Finally, we prove that a wave starting close to the
peakon remains close to some translate of it at all later times, that is, the
shapes of these peakons are stable under small perturbations.
