Let \(z_{k}(k=1, \cdots, n)\) be complex numbers lying on the same side of a straight line passing through the origin. Show that
\[ z_{1}+z_{2}+\cdots+z_{n} \neq 0, \quad 1 / z_{1}+1 / z_{2}+\cdots+1 / z_{n} \neq 0 \]
Hint: Consider a special situation first.
Solution
Rotate the plane so that the given line becomes the imaginary axis and all the \(z_k\) lie in the open right half-plane. Multiplication by a unimodular constant does not affect whether either displayed sum is zero, so it is enough to treat this case. Then \[ \Re z_k>0 \qquad(k=1,\dots,n). \] Hence \[ \Re\left(\sum_{k=1}^n z_k\right) =\sum_{k=1}^n\Re z_k>0, \] and therefore \(\sum z_k\ne0\).
Moreover, \[ \Re\frac1{z_k} =\Re\frac{\overline{z_k}}{|z_k|^2} =\frac{\Re z_k}{|z_k|^2}>0. \] Thus \[ \Re\left(\sum_{k=1}^n\frac1{z_k}\right)>0, \] so the sum of reciprocals is also nonzero.