Asymptotics of Gelfand's formula In the following paper, it is stated that for for any matrix norm, $n \in \mathbb{N}$ and $A \in \mathbb{C}^{d \times d}$, the following holds:
$\rho(A) \ge \gamma^{(1+\ln n)/n}\|A^{n}\|^{1/n}$
for some constant $\gamma \in (0,1)$ and $\rho(A)$ is the spectral radius of $A$. Unfortunately, I could not find the source of this claim. In what way does $\gamma$ depend on $d$ and the specific norm? I am interested in the $\infty$-norm, i.e., $\|A\| = \max_{i}\sum_{j=1}^{d}|A_{i,j}|$.
Thank you very much. 

Unless $\gamma$ also depends on $A$, I don't think the claim is true. With $n=1$, the claim boils down to $\rho(A) \ge \gamma \|A\|$ for all $A$, which is demonstrably false if you consider the maximum absolute row sum norm and $A=\pmatrix{1&x\\ 0&1}$. We have $\rho(A)=1$ but $\|A\|=|x|+1\to+\infty$ when $x\to\infty$.
