The product of the sum of two complex numbers is in the form $$(a+b)(a^*+b^*) = aa^* + bb^* + ab^* + a^*b$$ where $a,b \in \mathbb{C}$ and $aa^*\geq bb^*$.
Claim
Is this equation $$ab^* + a^*b \geq bb^*$$
always true or in what condition this can be true?
My effort
I have tried proving it but I can think of a number $b=i$ that leads me to
$$\mp \text{Im}\{2a\} \geq 1$$
which I suppose disproves the claim when $a$ has no imaginary component.
I am looking for a way to make the claim always true by adjusting the condition: $aa^*\geq bb^*$ or adding other condition.
Modifying the given condition from $|a| \geq |b|$ to $|a| \geq |a-b|$, the claim $$ab^* + a^*b \geq bb^*$$ is always true $a,b \in \mathbb{C}$.
Proof
From the condition, $$|a| \geq |a-b|$$ $$aa^* \geq (a-b)(a-b)^*$$ $$aa^* \geq (a-b)(a^*-b^*)$$ $$aa^* \geq aa^*-ab^*-a^*b+bb^*$$ $$0 \geq bb^*-ab^*-a^*b$$ $$ab^*+a^*b \geq bb^* \blacksquare$$