Starlikeness of analytic functions using special functions and subordination

Let \(\mathcal {P}\) $\mathcal{P}$ be the class of analytic functions having positive real part in the complex plane. The association of subordination and special functions is used to find sharp estimates on the parameter \(\beta \) $\beta$ such that the analytic function \(p\in \mathcal {P}\) $p\in \mathcal{P}$ is subordinate to certain functions having positive real part whenever \(p(z)+\beta z p'(z)\) $p(z)+\beta z{p}^{\prime }(z)$ is subordinate to the Janowski function. Further, the concept of admissibility is employed to establish certain second and third order differential subordination relations between the analytic function p and the functions associated with right half plane. As a sequel, we demonstrate the starlikeness of various well-known analytic functions as well.