For two complex numbers $c_1, c_2$ such that $|c_1|,|c_2|\leq 1$, is the following inequality true?
$$2|c_1|^2|c_2|^2 \leq c_1c_2^* + c_1^*c_2$$
Here $c^*$ is the complex conjugate of $c$. For real numbers, this is easily shown but I am not sure how to prove this for complex numbers. I have tried a few examples numerically and haven't found a counterexample so far.
It's not true. Pick two complex numbers $c_1, c_2$ such that $c_1 \overline{c_2}$ is a non-zero real multiple of $i$, e.g. $c_1 = i$ and $c_2 = 1$. Then $$c_1 \overline{c_2} + \overline{c_1} c_2 = 2\operatorname{Re}(c_1 \overline{c_2}) = 0.$$ If the inequality were true, then we would have $2|c_1|^2 |c_2|^2 \le 0$, which would imply that $c_1 = 0$ or $c_2 = 0$, and hence $c_1 \overline{c_2} = 0$, against assumption.