A problem regarding the in-equality of complex numbers .

Hint: in conjunction with the two given expressions, also consider the expressions $$ \bigg| \sum_{j=1}^n x_j + i \sum_{j=1}^n y_j \bigg| \quad\text{and}\quad \frac1{\sqrt2} \bigg( \sum_{j=1}^n x_j + \sum_{j=1}^n y_j \bigg). $$

Since it was somewhere tagged as duplicate here my answer:

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 $x_k,y_k\geq 0$.

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