Let $z_{1},\ldots z_{N}$ be complex numbers, and $a_{1},\ldots a_{N}$ be fixed complex numbers with positive real parts, such that $\arg a_{1}>\arg a_{2}>\ldots>\arg a_{N}$. Assume

$$\text{Re}(z_{i})=\text{Re}(a_{i}),\quad\text{Im}\sum_{1}^{m}z_{i}\geq\text{Im}\sum_{1}^{m}a_{i},\quad\sum_{1}^{N}z_{i}=\sum_{1}^{N}a_{i}.$$

Then $\sum_{1}^{N}|z_{i}|\geq\sum_{1}^{N}|a_{i}|$.