$D(-1)$-tuples in the ring $\mathbb{Z}[\sqrt{-k}]$ with $k＞0$

Let $n$ be a non-zero integer and $R$ a commutative ring. A $D(n)$-$m$-tuple in $R$ is a set of $m$ non-zero elements in $R$ such that the product of any two distinct elements plus $n$ is a perfect square in $R$. In this paper, we prove that there does not exist a $D(-1)$-quadruple $\{a,b,c,d\}$ in the ring $\mathbb{Z}[\sqrt{-k}]$, $k\ge 2$ with positive integers $a＜b\le8a-3$ and negative integers $c$ and $d$. By using that result, we were able to prove that such a $D(-1)$-pair $\{a,b\}$ cannot be extended to a $D(-1)$-quintuple $\{a,b,c,d,e\}$ in $\mathbb{Z}[\sqrt{-k}]$ with integers $c$, $d$ and $e$. Moreover, we apply the obtained result to the $D(-1)$-pair $\{p^i, q^j\}$ with arbitrary different primes $p$, $q$ and positive integers $i$, $j$.