We study the evolution of convex hypersurfaces with initial at a rate equal to <i>H</i> 鈥<i>f</i> along its outer normal, where <i>H</i> is the inverse of harmonic mean curvature of is a smooth, closed, and uniformly convex hypersurface. We find a <i>胃</i>* > 0 and a sufficient condition about the anisotropic function <i>f</i>, such that if <i>胃</i> > <i>胃</i>*, then remains uniformly convex and expands to infinity as <i>t</i> 鈫+ 鈭and its scaling, , converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log<i>H</i>-log <i>f</i> instead of <i>H</i>-f.