TITLE: Matrix exponential bound
QUESTION [5 upvotes]: I am looking to find analogues for products of matrices of the scalar inequalities
$$|1+x|\leq e^x,\qquad \big|\frac{1}{1+x}\big|\leq e^{-x+x^2},$$
which hold for $|x|\leq 1/2$.
Take $n,d\geq 1$, $A_1,\ldots,A_n\in\mathbb{R}^{d\times d}$ such that $\|A_i\|\leq 1/2$, where $\|\cdot\|$ stands for the operator norm.
Are the following inequalities true?
$$
\big\|\prod_{i=1}^n(I+A_i)\big\|\leq \big\|\exp\big(\sum_{i=1}^nA_i\big)\big\|
$$
$$
\big\|\prod_{i=1}^n(I+A_i)^{-1}\big\|\leq\big\|\exp\big(\sum_{i=1}^n-A_i+A_i^2\big)\big\|
$$

REPLY [4 votes]: No. When $A_1=-A_2=A=\frac12\pmatrix{0&-1\\ 1&0}$,
$$
\left\|(I+A)(I-A)\right\|=\|I-A^2\|=\left\|\frac54I\right\|=\frac54>1=\|\exp(A-A)\|
$$
and
\begin{align}
&\left\|(I+A)^{-1}(I-A)^{-1}\right\|=\left\|\frac45I\right\|=\frac45\\
>\ &0.6065=e^{-1/2}
=\|e^{-I/2}\|
=\|e^{2A^2}\|
=\left\|\exp\left(-A+A^2-(-A)+(-A)^2\right)\right\|.
\end{align}