Where is f(z)/f'(z) univalent?

2015/03/17 by Milutin Obradović, Obradović, Milutin, Saminathan Ponnusamy +3 · 1 citation
Materials Science · Mathematics · #30C45 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Polymer Synthesis and Characterization

paper · pdf · doi:10.48550/arxiv.1503.04931

openalex publication_date 2015/03/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30

Abstract

Let \mathcal S denote the family of all univalent functions f in the unit disk \ID with the normalization f(0)=0= f'(0)-1. There is an intimate relationship between the operator P_f(z)=f(z)/f'(z) and the Danikas-Ruscheweyh operator T_f:=∫₀^z(tf'(t)/f(t)) dt. In this paper we mainly consider the univalence problem of F=P_f, where f belongs to some subclasses of \mathcal S. Among several sharp results and non-sharp results, we also show that if f∈ \mathcal S, then F ∈ \mathcal U in the disk |z|

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