A holomorphic function F defined on the unit disc belongs to $A^{p,{alpha}}$ (0 < p < $infty$, 1 < ${alpha}$ < $infty$) if $intlimits_U|F(z)|^p frac{1}{1-|z|}(1+log)frac{1}{1-|z|})^{-alpha}$ dxdy < $infty$. For boundedness of the composition operator defined by $C_{fg}=g{circ}f$ mapping Blochs into $A^{p,{alpha}$ the following (1) is a sufficient condition while (2) is a necessary condition. (1) $intlimits_o^1frac{1}{1-r}(1+logfrac{1}{1-r})^{-alpha}M_p(r,lambda{circ}f)^p;dr$ < $infty$ (2) $intlimits_o^1frac{1}{1-r}(1+logfrac{1}{1-r})^{-alpha+p}(1-r)^pM_p(r,f^#)^p;dr$ < $infty$.