We are supposed to show that $$|ab^* + a^*b| \leq 2|ab|$$ where a and ba re complex numbers and a* and b* are their respective conjugates (so $a = x_1+iy_1$, $a^* = x_1-iy_1$, $b = x_2+iy_2$, $b^* = x_2-iy_2$)
I've gotten as far as $|2x_1x_2 +2y_1y_2| \leq 2|x_1x_2 + ix_1y_2 + iy_1x_2 - y_1y_2$
but when I try to take the magnitudes of these, the i's in the second term all become negative ($ i^2 = -1$) and then I end up with $2|(x_1x_2)^2 - (x_1y_2)^2 - (y_1x_2)^2 - (y_1y_2)^2|$ which seems like it has to be less than $|2x_1x_2 +2y_1y_2|$ not greater than.
I've been staring at this for a few hours and not making any progress so I'm sure I'm missing something, I'm just not sure what.
Thank you!
$$|a\bar b + \bar a b| \le |a\bar b| + |\bar a b| = |ab| + |ab| = 2|ab|$$ where the first step is the triangle inequality, and the second uses $|zw| = |z|\,|w|$ and $|\bar z| = |z|$.