Sharp starlikeness conditions for analytic functions with bounded derivative

We develop sharp conditions for various types of starlikeness for functions analytic in the unit disk with bounded derivatives. We also describe the p recise range (zf ' (z) /f (z) : z is an element of D, f is an element of I- lambda), where f is an element of I-lambda means f (o) = o f ' (o) = 1 and \f ' (z) - 1 \ less than or equal to lambda in the unit disc D, and draw so me conclusions from that.