First order differential subordination for functions with positive real part

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 $1+\beta z p’(z), ,,1+\beta z p’(z)/p(z), ,,\mbox{or},,1+\beta z p’(z)/p^{2}(z)$issubordinateto$\sqrt{1+z}$. Our results provide sharp version of previously known results.