A new semilocal convergence theorem for the Secant method under Hölder continuous divided differences

A new Kantorovich-type semilocal convergence theorem for the Secant method in Banach spaces is provided for approximating a solution of a nonlinear operator equation. It is assumed that the first-order divided difference of the nonlinear operator is Hölder continuous. Our convergence conditions, strategically proposed, differ from some existing ones and are easily satisfied. Therefore our results are of theoretical and practical interest. Finally, two simple examples are provided to show that our results apply, where earlier ones fail.