Some properties of certain expressions of analytic functions

Let AA denote the class of functions f(z)f(z) which are analytic in the open unit disk and normalized by f(0)=f′(0)−1=0f(0)=f′(0)−1=0. In this paper the expression f′(z)−1f(z)/z is studied using differential subordinations and different properties of f(z)z, as well as sufficient conditions for starlikeness and univalence of f(z)∈Af(z)∈A, are obtained. Also, several open problems are posed.