Application of Pythagorean means and Differential Subordination

Abstract

For $0\le \alpha \le 1,$ 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_α(p(z),p(z)Θ(z)+zp'(z)Φ(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.