On Convex Dominants of Exact Differential Subordination

TL;DR

This paper investigates convex dominant functions related to differential subordination involving specific admissible functions, establishing the existence of the best dominant, deriving an exact differential equation, and providing bounds and criteria for univalence.

Contribution

It introduces new admissible functions for differential subordination, finds the best convex dominant, and establishes univalence criteria with sharp bounds.

Findings

Identified the best convex univalent dominant function for the differential subordination.

Proved that the differential subordination leads to an exact differential equation.

Derived sharp lower bounds for the real part of p and univalence criteria.

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 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.