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? $$ \big\|\prod_{i=1}^n(I+A_i)\big\|\leq \big\|\exp\big(\sum_{i=1}^nA_i\big)\big\| $$ $$ \big\|\prod_{i=1}^n(I+A_i)^{-1}\big\|\leq\big\|\exp\big(\sum_{i=1}^n-A_i+A_i^2\big)\big\| $$
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}