Let be the complex unit disc and the class of functions . For also satisfing the condition
for suitable , , and , the authors give sufficient conditions so that the function defined by
(with chosen so that the above formula generalizes some results of other authors, but also provide new results) is starlike. Particular cases of are taken into account. Some consequences are also given. One of them gives a sharp estimate for the real constant that ensures starlikeness of a function that satisfies the condition .