Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric)
$$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ & & \ddots\end{pmatrix}$$
and suppose that it's an M-matrix, diagonally dominant, with $a_i,b_i,c_i>0$, $a_i\ge c_{i-1}+ b_i$, ($c_0=b_n=0$) and such that there exists at least an index $i$ with $a_i> c_{i-1}+ b_i$.
I'm interested in an upper bound for the infinity norm of the inverse.
I already know that if $0<\gamma\le a_i-c_{i-1}- b_i$ for all indexes, then $$\|L^{-1}\|_{\infty}\le \gamma^{-1}$$ and, more in general, if $w\ge 0$ and $Lw\ge \gamma e$, then $$\|L^{-1}\|_{\infty}\le \gamma^{-1}\|w\|_{\infty}$$
I also know that for the particular matrix with $a_i=2, b_i=c_i=1$, we have $$\|L^{-1}\|_{\infty}\le \frac{(n+1)^2}{8}$$
Is there some general result of this kind? And if the matrix is Toeplitz, there's a rule for the dependence of such a bound to $n^k$ for some $k$?
Note: This Link may be useful.