Applications of subordination theory to starlike functions

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎Atma Ram Sanatan Dharma College‎, ‎University of Delhi‎, ‎Delhi 110021‎, ‎India.

2 Department of Mathematics‎, ‎University of Delhi‎, ‎Delhi--110007‎, ‎India.

Abstract

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.

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