A derivative-free transformation preserving the order of convergence of iteration methods in case of multiple zeros

SummaryIfs is a zero of the functionf(x) with multiplicityn, and |f(n+1)(x)| is bounded in a neighborhood ofs, thens is a simple zero of the function $$h\left( x \right): = f^2 \left( x \right)/\left( {f\left( {x + f\left( x \right)} \right) - f\left( x \right)} \right)$$ and |h″(x)| is bounded in some neighborhood ofs. Iteration methods like Newton's Steffensen's or the secant method converge therefore tos without reduction of their order of convergence, if they are applied onh(x) instead off(x).