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

Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.}, one-to-one) functions $f(z)=z+{\sum }_{n=2}^{\mathrm{\infty }}{a}_n{z}^n$ in the unit disk $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$. For $f\in \mathcal{S}$, Ma proposed the generalized Zalcman conjecture that $$\mathrm{\mid }{a}_n{a}_m-a_{n+m-1}\mathrm{\mid }\le (n-1)(m-1),forn\ge 2,m\ge 2,$$ with equality only for the Koebe function $k(z)=z\mathrm{/}(1-z{)}^2$ and its rotations. In this paper using the properties of holomorphic motion and Krushkal’s Surgery Lemma \cite{Krushkal-1995}, we prove the generalized Zalcman conjecture when $n=2$, $m=3$ and $n=2$, $m=4$.