# Lower&upper bound $\alpha_1\|x_1\|_{\infty}+\alpha_2\|x_2\|_{\infty} \leq \|x_1+x_2\|_{\infty}\leq\beta_1\| x_1\|_{\infty}+\beta_2\| x_2\|_{\infty}$?

If $$x_1, x_2 \in \mathbb{C}^n$$, then is there any lower and upper bound for $$\|x_1 + x_2\|_{\infty}$$, where $$\| x\|_{\infty} := \max_{i=1,\ldots,n} |x_i|$$?

More specifically, I am wondering, is there a lower and upper bound in this below format $$\alpha_1 \|x_1\|_{\infty} + \alpha_2 \|x_2\|_{\infty} \leq \|x_1 + x_2\|_{\infty} \leq \beta_1 \|x_1\|_{\infty} + \beta_2 \|x_2\|_{\infty}$$? If yes, then what can be $$\alpha_1 \in \mathbb{R}$$, $$\alpha_2 \in \mathbb{R}$$, $$\beta_1 \in \mathbb{R}$$, and $$\beta_2 \in \mathbb{R}$$?

EDIT: For upper bound, the triangle inequality can be invoked, i.e., $$\|x_1 + x_2\|_{\infty} \leq \beta_1 \|x_1\|_{\infty} + \beta_2 \|x_2\|_{\infty}$$, where $$\beta_1 = 1$$ and $$\beta_2 = 1$$.

EDIT: For lower bound, one stupid option can be taking the difference such that $$\| x_1\|_{\infty} - \| x_2\|_{\infty} \leq \| x_1 + x_2\|_{\infty}$$ assuming $$\| x_1\|_{\infty} \geq \| x_2\|_{\infty}$$. Not sure if this will result into the tight bounds

• Are you familiar with the triangle inequality? May 15 at 13:56
• @TMGallagher Yes, true. That can give me an upper bound. How about the lower bound? May 15 at 13:58
• What have you tried toward seeing if a non-zero lower bound is possible? May 15 at 14:00
• @TMGallagher One stupid option can be taking the difference such that $\| x_1\|_{\infty} - \| x_2\|_{\infty} \leq \| x_1 + x_2\|_{\infty}$ assuming $\| x_1\|_{\infty} \geq \| x_2\|_{\infty}$. Not sure if this will result into the tight bounds May 15 at 14:04
• I definitely wouldn't call that stupid,'' and you can investigate whether it's a tight lower bound. May 15 at 14:16