On Convex Dominants of Exact Differential Subordination

Abstract

Let $h$ be a non vanishing convex univalent function and $p$ be an analytic function in $\mathbb{D}$. We consider the differential subordination $${\psi }_i(p(z),z{p}^{\prime }(z))\prec h(z)$$ with the admissible functions in consideration as ${\psi }_1:=(\beta p(z)+\gamma {)}^{-\alpha }\left(\frac{(\beta p(z)+\gamma )}{\beta (1-\alpha )}+z{p}^{\prime }(z)\right)$ and ${\psi }_2:=\frac{1}{\sqrt{\gamma \beta }}\arctan \left(\sqrt{\frac{\beta }{\gamma }}{p}^{1-\alpha }(z)\right)+\left(\frac{1-\alpha }{\beta p^{2(1-\alpha )}(z)+\gamma }\right)\frac{z{p}^{\prime }(z)}{p^{\alpha }(z)}$. 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 ${\psi }_i(q,nz{q}^{\prime }(z))=h(z)$. Further, we show that ${\psi }_i(q,z{q}^{\prime }(z))=h(z)$ 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.