C^1 linearization is of special interests because it can distinguish characteristic directions of dynamical systems. It is known that planar C^{1,\alpha} contractions with a fixed point at the origin admit C^{1,\beta} linearization with sufficiently small \beta>0 if \alpha=1 and admit C^{1,\alpha} linearization if (log |\lambda_1|/ log |\lambda _2|)-1 < \alpha<="1," where \lambda_1 and \lambda _2 are eigenvalues of the linear parts of the contractions with 0 < |\lambda _1|<="|\lambda" _2|< 1. in this talk we present some advances that the lower bound of \alpha is improved and the obtained h?lder exponent \beta of the linearization is sharp.