On Convex Dominants of Exact Differential Subordination

Let $h$ be a non vanishing convex univalent function and $p$ be an analytic function in $\mathbb{D}$. We consider the differential subordination $$\psi_i(p(z), z p'(z)) \prec h(z)$$ with the admissible functions in consideration as $\psi_1:=(\beta p(z)+\gamma)^{-\alpha}\left(\tfrac{(\beta p(z)+\gamma)}{\beta(1-\alpha)}+ z p'(z)\right)$ and $\psi_2:=\tfrac{1}{\sqrt{\gamma \beta}}\arctan\left(\sqrt{\tfrac{\beta}{\gamma}}p^{1-\alpha}(z)\right)+\left(\tfrac{1-\alpha}{\beta p^{2 (1-\alpha)}(z)+\gamma}\right)\tfrac{z p'(z)}{p^{\alpha}(z)}$. The objective of this paper is to find the dominants, preferably the best dominant(say $q$) of the solution of the above differential subordination satisfying $\psi_i(q, n zq'(z))= h(z)$. Further, we show that $\psi_i(q,zq'(z))= h(z)$ is an exact differential equation and $q$ is a convex univalent function in $\mathbb{D}$. In addition, we estimate the sharp lower bound of $\RE p$ for different choices of $h$ and derive a univalence criteria for functions in $\mathcal{H}$(class of analytic normalized functions) as an application to our results.