Journal of the
Korean Mathematical Society JKMS

Let $p$ be an analytic function defined on the open unit disk $\mathbb{D}$. We obtain certain differential subordination implications such as $\psi(p):=p^{\lambda}(z)(\alpha+\beta p(z)+\gamma/p(z)+\delta z p'(z)/p^{j}(z)) \prec h(z)$ $(j=1,2)$ implies $p \prec q$, where $h$ is given by $\psi(q)$ and $q$ belongs to $\mathcal{P}$, by finding the conditions on $\alpha$, $\beta$, $\gamma$, $\delta$ and $\lambda$. Further as an application of our derived results, we obtain sufficient conditions for normalized analytic function $f$ to belong to various subclasses of starlike functions, or to satisfy $|\log(z f'(z)/f(z))|<1$, $|(z f'(z)/f(z))^2-1|<1$ and $z f'(z)/f(z) $ lying in the parabolic region $v^2 <2u-1$.

Keywords: Carath\'{e}odory function, differential subordinations, minimum principle, exponential function, strongly starlike function, lemniscate of Bernoulli, Janowski starlike function

MSC numbers: 30C45, 30C80

Supported by: The work presented here was supported by a Research Fellowship from the Department of Science and Technology, New Delhi.