Inequalities concerning starlike functions and their n-th root

If A is the class of all analytic functions in the complex unit disc $\Delta$, of the form:$f(z) = z + a_2z^2 + \cdots$ and if $f \in A$ satisfies in $\Delta$ the condition:$$Re \frac{zf'(z)}{f(z)} > |\frac{zf'(z)}{f(z)}-1|$$ then Re \sqrt[n]{f(z)/z \geq (n+1)/(n+2)}. We show also that if $f$ is starlikein $\Delta$ (i.e. Re zf'(z)/f(z) > 0 in $\Delta$), then Re \sqrt[n]{f(z)/z > n(n+2)}.