  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 the
generalized 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 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
\cite{Krushkal-1995}, we prove the generalized Zalcman conjecture when $n=2$,
$m=3$ and $n=2$, $m=4$.
