The paper is concerned with solving nonlinear operator equations in Banach spaces by the Newton-Kantorovich method, and with improving the convergence conditions of this iterative method. Under the same hypotheses like in previous papers and with the same computational cost, a larger radius of convergence and a finer complexity are obtained. In addition to convergence, the author also attempts to answer to the fundamental question: What stronger conditions must be imposed to ensure an optimal complexity? The computational complexity of the Newton method is the total cost of computing the \(k\)-th iteration. The author defines the optimal complexity and derives estimates for the complexity and for optimal complexity of the method. Numerical examples justify the theoretical results and compare favorably with previous results.