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Let $\mathcal{S}$ denote the class of analytic and univalent ({\it i.e.},one-to-one) functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk$\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$. For $f\in \mathcal{S}$, Ma proposed thegeneralized Zalcman conjecture that $$|a_{n}a_{m}-a_{n+m-1}|\le(n-1)(m-1),\,\,\,\mbox{ for } n\ge2,\, m\ge 2,$$ with equality only for theKoebe function $k(z) = z/(1 - z)^2$ and its rotations. In this paper using theproperties of holomorphic motion and Krushkal's Surgery Lemma\cite{Krushkal-1995}, we prove the generalized Zalcman conjecture when $n=2$,$m=3$ and $n=2$, $m=4$.

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