How to prove the following matrices inequality?

Question

Let $A$ and $B$ be two matrices $(nxn)$. How can we prove the following inequality?

-->An OPINION: One friend disserts it is false. His counter-example is:

$n=4$, $B=2I_4$, $A=I_4$, and $\|M\|=\max_{\|x\|_2=1} \frac{\|Mx\|_2}{\|x\|_2}$. Then $(A+B)^n-A^n=(3^n-1) I_4$. So, $\|(A+B)^n-A^n\|=3^4-1$.

And $\|B\|\|A\|^{n-1}e^{\frac{\|B\|}{\|A\|}}=2e^2<3^4-1$.

What do you think?

Answer 1

Your friend is right. His counter-example works.