In $1869$, Darboux proved that, when using Newton method, there will never be an overshoot of the solution if the starting point $x_0$ is such that $$f(x_0) \times f''(x_0) >0$$ and, from a numerical point of view, this is crucial.
I am wondering if similar conditions (even if complex) have been established for higher order methods (Halley, Householder, ...).
I searched in vain; so my question.