On Brownʹs and Newtonʹs methods with convexity hypotheses

In the context of the monotone Newton theorem (MNT) it has been conjectured that discretised Brown iterations converge at least as fast as discretised Newton iterations, because such is the case for analytic iterations. With easily verified hypotheses, it is proved here that Brown analytic iterations converge strictly faster than Newton ones. As a consequence, the same result holds for discretised iterations with conveniently small incremental steps. However, in the general context of the MNT, it may happen that Newtonʹs discretised method converges faster than Brownʹs, but this situation can be remedied in many cases by conveniently shifting the initial value, so that those hypotheses ensuring the reverse are satisfied. Thus, a fairly effective solution is given to the problem stated initially.