A sharp criterion for the univalence of the Libera operator

Let f be an analytic function of the form f(z)= z + a2z + … de?ned in the unit disc U = {z ? C : |z| < 1}. Suitable values of ? have been determined by a number of authors, so that Re(zf (x)+ z2 3 f (z)  > ??, z ? U implies the starlikeness of f. In all these previous papers the method of differential subordination has been used. We improve their results using the method of convolution and obtain the biggest possible value of ? so that the differential inequality Re(zf (x)+ z2 3 f (z)  > ??, z ? U implies the univalence of the function f. The integral version of the result involving Libera operator is given.