Region of variability for certain subclass of univalent functions

2022/09/21 by Jnana Preeti Parlapalli, Vasudevarao Allu, Parlapalli, Jnana Preeti +1
Mathematics · #30C45 #30C50 #30C80 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2209.10424

openalex publication_date 2022/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathbbD:=\z∈ ℂ: |z|<1\ be the unit disk. For 0<α<1, let f_α(z)=z/(1-z^α) for z ∈ \mathbbD. We consider the class F of analytic functions f_α which satisfy \Re (1+zf"_α(z)/f'_α(z)) > β for 0<β<1. In this paper, we determine the region of variability of log f'_α(z₀) for fixed z₀ ∈ \mathbbD when f varies over the class \mathcal F(λ):=\f_α ∈ F: f_α(0)=0, f'_α(0)=1 and f"_α(0)=2λ(1-β) for 0≤ λ≤ 1\.

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