Some simple sufficient conditions for starlikeness and convexity

Let A be the class of analytic functions in the unit disc that are of the form f ( z ) = z + a 2 z 2 + ⋯ . In this work the expressions f ′ ( z ) − ( 1 − γ ) f ( z ) / z and z f ″ ( z ) − γ f ′ ( z ) , 0 < γ ≤ 1 , are studied, as sufficient conditions for starlikeness and convexity.