Sharp estimates on $\beta$ are determined so that an analytic function $p$ defined on the open unit disk in the complex plane normalized by $p(0)=1$ is subordinate to some well known starlike functions with positive real part whenever $1+\beta z p'(z), \,\,1+\beta z p'(z)/p(z), \,\,\mbox{ or}\,\,1+\beta z p'(z)/p^{ 2}(z)$ is subordinate to $\sqrt{ 1+z}$. Our results provide sharp version of previously known results.
Categories: math.CV