Differential subordination and harmonic means for the Koebe type $n$-fold symmetric functions

Year 2022, Volume: 51 Issue: 6, 1630 - 1636, 01.12.2022

Syed Zakar Hussain Bukhari , Maryam Nazir

https://doi.org/10.15672/hujms.1072813

Abstract

In this research, we develop some differential subordination results involving harmonic means of $f_{b}(z),f_{b}(z)+zf_{b}^{\prime}(z)$ and $f_{b}(z)+\frac{zf_{b}^{\prime}(z)}{f_{b}(z)},$ where $f_{b}(z)=\frac{z}{\left(1-z^{n}\right) ^{b}},$ $b\geq0;n\in\mathbb{N}=1,2,3...$ is an $n$-fold symmetric Koebe type functions defined in the unit disk with $f_{b}(0)=0,f_{b}^{\prime}(z)\neq0.$ By using the admissibility conditions, we also study several applications in the geometric function theory.

Keywords

Differential subordination, harmonic means, Koebe type n-fold symmetric function

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