2025/11/12 by Ahamed, Molla Basir, Roy, Partha Pratim
#30B10 #30C35 #30C62 #30H05 #31A05 #Complex Variables (math.CV) #FOS: Mathematics #Primary 30A10 #Secondary 30C45
paper · doi:10.48550/arxiv.2511.09121
In this paper, we study the class Σ(m)(p) of meromorphic univalent functions f in \mathbbD with a pole of order m ≥ 1 at p ∈ (0,1), admitting a k-quasiconformal extension (0 ≤ k < 1) to \widehatℂ. Using the Area Theorem and convolution methods, we establish a generalized area-type inequality and derive explicit analytic membership conditions for Σ(m)(p). We also extend the convolution theorem to a modified Hadamard product of m functions, fj ∈ Σ(m)kj(p), determining sufficient conditions for the product to be in Σ(m)α(p), with α defined by kj and p. Further results include a sufficient criterion for sense-preserving harmonic mappings on convex domains to admit quasiconformal extensions, and the sharp Schwarzian norm for f ∈ Σk(p) (the m=1 case). These findings improve upon existing results of [\em Proc. Amer. Math. Soc., 144(6) (2016), 2593--2601].