On criteria for starlikeness and convexity of analytic functions

In this paper the authors study (1 - gamma + zf" (z) /f'(z) / f'(z) / zf'(z) / f(z)) as a criteria for starlikeness and convexity. Sharp upper bound of |a(2)| and of the Fekete-Szego functional |a(3)-mua(2)(2)| is given for a class of analytic functions defined by using this expression.