  This paper is concerned with the monotonicity of the period function for
closed orbits of systems of the Li\'enard II type equation given by $\ddot{x} +
f(x)\dot{x}^{2} + g(x) = 0$. We generalize Chicone's result regarding the
monotonicity of the period function to planar Hamiltonian vector fields in the
presence of a position dependent mass. Sufficient conditions are also given for
the isochronicity of the potential in case of such a system.
