Once we have seen the influence of convexity on the speed of convergence of the Newton method in Chap. 2 , we now give three acceleration procedures for the Newton method, which are based on the influence of convexity and lead us to the three well-known third-order iterative methods of Chebyshev, Super-Halley, and Halley. Next, we study these three methods in the scalar case and in Banach spaces. In the scalar case, we see the influence that convexity has on each of the three methods, and from this, we guarantee that we can always solve a scalar equation by any of the three methods under certain restrictions on the degree of logarithmic convexity of the function involved in the equation to be solved. The study of these three methods in Banach spaces is carried out from the same point of view: the method of majoring sequences developed by Ortega for the Newton method, which simplifies the seminal majorant principle developed by Kantorovich to analyze the semilocal convergence of the Newton method in Banach spaces. We illustrate all theoretical results with examples, and, as in Chap. 2 for the Newton method, we use the theoretical significance of the methods to draw conclusions about the existence of solution of the equation to be solved.