  We consider the modified Zakharov-Kuznetsov (mZK) equation in two space
dimensions in both focusing and defocusing cases. Using the $I$-method, we
prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for
any data in the defocusing case and under the assumption that the mass of the
initial data is less than the mass of the ground state solution of $\Delta
\varphi - \varphi + \varphi^3 = 0$ in the focusing case. This improves the
global well-posedness result of Linares and Pastor [20].
