Application of Pythagorean means and Differential Subordination

TL;DR

This paper explores differential subordination involving Pythagorean means and their combinations, providing new implications and conditions that ensure functions are starlike or univalent, thus generalizing existing mathematical results.

Contribution

It introduces novel differential subordination implications using Pythagorean means and combines classical means, extending the theory and applications in geometric function theory.

Findings

Established new differential subordination implications involving harmonic means.

Generalized existing results to broader classes of functions.

Provided sufficient conditions for starlikeness and univalence.

Abstract

For $0\leq\alpha\leq 1,$ let $H_{\alpha}(x,y)$ be the convex weighted harmonic mean of $x$ and $y.$ We establish differential subordination implications of the form \begin{equation*} H_{\alpha}(p(z),p(z)\Theta(z)+zp'(z)\Phi(z))\prec h(z)\Rightarrow p(z)\prec h(z), \end{equation*} where $\Phi,\;\Theta$ are analytic functions and $h$ is a univalent function satisfying some special properties. Further, we prove differential subordination implications involving a combination of three classical means. As an application, we generalize many existing results and obtain sufficient conditions for starlikeness and univalence.