In this paper, we introduce the concept of ternary antiderivation on ternary Banach algebras and investigate the stability of ternary antiderivation in ternary Banach algebras, associated to the $(\alpha,\beta)$-functional inequality: \begin{align*} &\Vert \mathcal{F}(x+y+z)-\mathcal{F}(x+z)-\mathcal{F}(y-x+z)-\mathcal{F}(x-z)\Vert \nonumber\\ &\leq \Vert \alpha (\mathcal{F}(x+y-z)+\mathcal{F}(x-z)-\mathcal{F}(y))\Vert + \Vert \beta (\mathcal{F}(x-z)\\ &+\mathcal{F}(x)-\mathcal{F}(z))\Vert \end{align*} where $\alpha$ and $\beta$ are fixed nonzero complex numbers with $\vert\alpha \vert +\vert \beta \vert<2$ by using the fixed point method.