Some simple sufficient conditions for starlikeness and convexity

Let AA be the class of analytic functions in the unit disc that are of the form f(z)=z+a2z2+⋯f(z)=z+a2z2+⋯. In this work the expressions f′(z)−(1−γ)f(z)/zf′(z)−(1−γ)f(z)/z and zf″(z)−γf′(z),0<γ≤1zf″(z)−γf′(z),0<γ≤1, are studied, as sufficient conditions for starlikeness and convexity.