2017/09/02 by Vibhuti Arora, Arora, Vibhuti, Swadesh Kumar Sahoo +1 · 1 citation
Mathematics · #30C45 #30C55 #30D30 #34M05 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:30C45 #msc:30C55 #msc:30D30 #msc:34M05
paper · pdf · doi:10.48550/arxiv.1709.00529
16 pages. Submitted to a journal
arxiv created 2017/09/02 · openalex publication_date 2017/09/02 · arxiv updated 2017/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the family of all meromorphic functions f of the form f(z)=(1)/(z)+b0+b1z+b2z2+⋯ analytic and locally univalent in the puncture disk \mathbbD0:=\z∈ℂ: 0<|z|<1\. Our first objective in this paper is to find a sufficient condition for f to be meromorphically convex of order α, 0≤ α<1, in terms of the fact that the absolute value of the well-known Schwarzian derivative Sf (z) of f is bounded above by a smallest positive root of a non-linear equation. Secondly, we consider a family of functions g of the form g(z)=z+a2z2+a3z3+⋯ analytic and locally univalent in the open unit disk \mathbbD:=\z∈ℂ: |z|<1\, and show that g is belonging to a family of functions convex in one direction if |Sg(z)| is bounded above by a small positive constant depending on the second coefficient a2. In particular, we show that such functions g are also contained in the starlike and close-to-convex family.