An Inequality about complex numbers with non-negative real and imaginary parts

Question

Let $z_1,z_2 \dots dots z_n \in \mathbb{C} $ be such that the real and imaginary parts of each z_i are non-negative. Show that

$\Big|\sum_{i=1}^{n} z_i\Big| \geq\frac{1}{\sqrt{2}}\sum_{i=1}^{n} |z_i|$

I couldn't start this problem. I tried to get a geometric view of this but that didn't help

Answer 1

You may use

• $\sqrt{a+b}\leq \sqrt a + \sqrt b$ for $a,b \geq 0$ and
• $(x+y)^2 \leq 2(x^2+y^2)$ (This is $2$-dimensional Cauchy-Schwarz inequality)

Let $z_k = x_k + iy_k$ for $k=1, \ldots , n$ where $x_k,y_k\geq 0$.

$$\left(\sum_{k=1}^n \lvert z_k \rvert\right)^2 =\left(\sum_{k=1}^n \sqrt{x_k^2 + y_k^2}\right)^2 \leq \left(\sum_{k=1}^n \left(x_k + y_k\right)\right)^2$$ $$= \left(\sum_{k=1}^n x_k + \sum_{k=1}^n y_k\right)^2$$ $$\leq 2\left(\left(\sum_{k=1}^n x_k\right)^2 + \left(\sum_{k=1}^n y_k\right)^2\right) = 2 \lvert \sum_{k=1}^n z_k \rvert^2$$