On Certain Exact Differential Subordinations Involving Convex Dominants

Abstract

Let h be a non-vanishing convex univalent function and p be an analytic function in $\mathbb {D}$. We consider the differential subordination $\psi _i(p(z), z p'(z)) \prec h(z)$ with the admissible functions $\psi _1:=(\beta p(z)+\gamma )^{-\alpha }\left( \tfrac{(\beta p(z)+\gamma )}{\beta (1-\alpha )}+ z p'(z)\right)$ 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 $\psi _i(q(z), n zq'(z))= h(z)$. Furthermore, we show that $\psi _i(q(z),zq'(z))= h(z)$ is an exact differential equation and q is a convex univalent function in $\mathbb {D}$. In addition, we estimate the sharp lower bound of ${{\,\mathrm{Re}\,}}p$ for different choices of h and derive a univalence criterion for functions in $\mathcal {H}$ (class of analytic normalized functions) as an application to our results.