Univalence and Starlikeness of Certain Classes of Analytic Functions

For the analytic functions phi(zeta) = zeta + Sigma(infinity)(k=n) phi k zeta(k) in the unit disk O, we calculate the values of n and a, where the condition R(1 + zeta phi ''(zeta)/ phi'(zeta)) > -alpha or R <(1 + zeta phi ''(zeta)/phi'/(zeta)) < 1 + alpha/2 yields univalence and starlikeness. Conditions imply f in the class where all normalized analytic functions U, with vertical bar(zeta/phi(zeta))(2) phi'(zeta) - 1 vertical bar < 1 are obtained. Recent findings are gained, and unique cases are demonstrated. The generalization of the Jack lemma serves the proof of the main result and that our methodology is based on the idea of subordination.