The abc-conjecture consequence for three-term monomial differences

Let $a,b,c\in\mathbb{N}$ be fixed natural numbers and let $n_{1},n_{2},n_{3}\in\mathbb{N}$. Denote

$$D=n_{1}^{a}-n_{2}^{b}n_{3}^{c}.$$

The abc-conjecture consequence. If $D\neq 0$, then one has the lower bound

$$\lvert D\rvert\gg n_{1}^{a-1-\varepsilon}n_{2}^{-1}n_{3}^{-1}.$$

This estimate is presented as a consequence of the abc-conjecture and is used to sharpen the error term in an asymptotic evaluation of a triple zeta integral. Its resolution status is not specified in the source; as stated, it depends on the unresolved abc-conjecture.