The objective of this study is to investigate the Contraharmonic-mean Newton’s method (CHN) for solving equations in Banach space. Local analysis has been performed by Taylor’s expansion utilizing high order derivatives and also using only the first derivatives appearing on the method. We have checked the theoretical results by comparing the CHN method with the Newton’s and related third order methods by Weerakoon et al. utilizing some test functions. The theoretical and numerical results are examined by the dynamical analysis of a selected test function.