202212301726 Solution to 1987-CE-AMATH-I-3

For any complex number $z$ , let $\overline{z}$ , $|z|$ , and $\mathrm{Re}(z)$ be its conjugate, modulus, and real part respectively. Show that

$z+\overline{z}=2\mathrm{Re}(z)$ and $|z|\geqslant \mathrm{Re}(z)$ .

Hence, or otherwise, show that for any complex numbers $z_1$ and $z_2$ ,

$z_1z_2+\overline{z_1z_2}\leqslant 2|z_1||z_2|$ .

Roughwork.

In the field $\mathbb{C}=\{ x+iy:x,y\in\mathbb{R}\}$ of complex numbers, $x$ is called the real part of $z$ and $y$ the imaginary part, i.e.,

$z=x+iy=\mathrm{Re}(z)+i\,\mathrm{Im}(z)$ ,

its trigonometric form being $z=r(\cos\theta +i\sin\theta )$ with $r$ the modulus and $\theta$ the argument, and its exponential form, $z=re^{i\theta}$ .

This problem is not to be attempted.

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