The problem I cause is attached below. I am trying to prove the inequality.
By using small $M$, I found that the terms on the left side of the inequality are part of the terms of the expansion on the right side. But I always feel dizzy when I try to prove the inequality by expanding the left side and the right side and then comparing them...
I am thinking that is there any properties for the power of strictly triangular matrix, or the matrix L1 norm can be used to the proof? Thank you in advance.
$ Q_{M}=\left[\begin{array}{ccccc} 0\\ k_{1} & 0\\ \frac{1}{2}k_{2} & \frac{1}{2}k_{1} & 0\\ \vdots & & \ddots & \ddots\\ \frac{1}{M-1}k_{M-1} & \frac{1}{M-1}k_{M-2} & \cdots & \frac{1}{M-1}k_{1} & 0 \end{array}\right] $
Prove that for $\forall n,M\in\mathbb{N}_{+}$,
$ n!\left\Vert Q_{M}^{n}\right\Vert _{1}\leq\left(k_{1}+\frac{1}{2}k_{2}+\cdots\frac{1}{M-1}k_{M-1}\right)^{n} $