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

@inproceedings{Allu2022ProofOT, title={Proof of The Generalized Zalcman Conjecture for Initial Coefficients of Univalent Functions}, author={Vasudevarao Allu and Abhishek Pandey}, year={2022} }

. Let S denote the class of analytic and univalent ( i.e. , one-to-one) functions f ( z ) = z + P ∞ n =2 a n z n in the unit disk D = { z ∈ C : | z | < 1 } . For f ∈ S , Ma proposed the generalized Zalcman conjecture that | a n a 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 [12], we prove the generalized Zalcman…

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