We provide an improved compared to <img border=0 width=45 height=24 src="http://fbpe/img/cubo/v12n1/img31.jpg">local convergence analysis and complexity for the interpolatory Newton method for solving equations in a Banach space setting. The results are obtained using more precise error bounds than before <img border=0 width=45 height=24 src="http://fbpe/img/cubo/v12n1/img31.jpg">and the same hypotheses/computational cost.