%0 Journal Article %T Applications of subordination theory to starlike functions %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Sharma, K. %A Ravichandran, V. %D 2016 %\ 06/01/2016 %V 42 %N 3 %P 761-777 %! Applications of subordination theory to starlike functions %K convex and starlike functions %K differential subordination %K univalent functions %R %X Let $p$ be an analytic function defined on the open unit disc $\mathbb{D}$ with $p(0)=1.$ The conditions on $\alpha$ and $\beta$ are derived for $p(z)$ to be subordinate to $1+4z/3+2z^{2}/3=:\varphi_{C}(z)$ when $(1-\alpha)p(z)+\alpha p^{2}(z)+\beta zp'(z)/p(z)$ is subordinate to $e^{z}$. Similar problems were investigated for $p(z)$ to lie in a region bounded by lemniscate of Bernoulli $|w^{2}-1|=1$ when the functions $(1-\alpha)p(z)+\alpha p^{2}(z)+\beta zp'(z)$ , $(1-\alpha)p(z)+\alpha p^{2}(z)+\beta zp'(z)/p(z)$ or $p(z)+\beta zp'(z)/p^{2}(z)$ is subordinate to $\varphi_{C}(z)$. Related results for $p$ to be in the parabolic region bounded by the $RE w=|w-1|$ are investigated. %U http://bims.iranjournals.ir/article_811_b56bdeb24d06f65b35a6ba3a70fd9fd6.pdf