  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.
