I have to prove for $A \in \mathbb{C}^{n\times n}$ the following statement:
If denominator on the right side of one of the two following statments is positive, then: $$ \left \| A^{-1} \right \|_{\infty } \leq 1/ \min_{i}(\left | A_{ii} \right | - \sum_{k\neq i} \left | A_{ik} \right |) $$ $$ \left \| A^{-1} \right \|_{1} \leq 1/ \min_{k}(\left | A_{kk} \right | - \sum_{i\neq k} \left | A_{ik} \right |)$$
We have to prove that for $y=Ax$, we have $\|y\|_\infty\le 1\implies\|A^{-1}y\|_\infty\le \dfrac1{\min_i(|A_{ii}|-\sum_{k\ne i}|A_{ik}|)}$, i.e., $$\|Ax\|_\infty\,\le\, \min_i(|A_{ii}|-\sum_{k\ne i}|A_{ik}|)\ \implies\ \|x\|_\infty\le 1\\ \forall i,j:|(Ax)_j|\,\le\, |A_{ii}|-\sum_{k\ne i}|A_{ik}|\ \implies\ \forall i:|x_i|\le 1$$ Now to the contrary, assume that the highest absolute value coordinate $x_i$ of $x$ has $|x_i|>1$, then $$|(Ax)_i|=\left|\sum_kA_{ik}x_k\right| \, =\,\left|A_{ii}x_i -\sum_{k\ne i}A_{ik}(-x_k)\right|\ \ge \\ \ge\ |A_{ii}||x_i|\,-\,\sum_{k\ne i}|A_{ik}||x_k|\ \ge\ |x_i|\cdot\left(|A_{ii}| - \sum_{k\ne i}|A_{ik}|\right)\ >\ \left(|A_{ii}| - \sum_{k\ne i}|A_{ik}|\right)$$ contradicting the condition.
$ $
Can be done in a similar manner.