Proof of The Generalized Zalcman Conjecture for Initial Coefficients of Univalent Functions

2022/09/21 by Vasudevarao Allu, Allu, Vasudevarao, Abhishek Pandey +1
Mathematics · #30C45 #30C50 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2209.11231

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

Abstract

Let S denote the class of analytic and univalent (\it i.e., one-to-one) functions f(z)= z+∑_n=2^∞a_n zⁿ in the unit disk \mathbbD=\z∈ ℂ:|z|<1\. For f∈ S, Ma proposed the generalized Zalcman conjecture that |a_na_m-a_n+m-1|≤ (n-1)(m-1), for n≥2, m≥ 2, with equality only for the Koebe function k(z) = z/(1 - z)² and its rotations. In this paper using the properties of holomorphic motion and Krushkal's Surgery Lemma \citeKrushkal-1995, we prove the generalized Zalcman conjecture when n=2, m=3 and n=2, m=4.

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