Let denote the class of functions , , that are analytic in the unit disk and let be the class of starlike functions and the class of convex functions in . is the integral operator defined by
Note that if . Using the theory of differential subordinations, the author proves some criteria involving or only, in determining the starlikeness or convexity of or of . For , , the conditions imposed to or are of the form , or , where is a suitable constant, depending on (and for ). Some interesting examples, which point out the usefulness of the new starlikeness and convexity criteria, are given.