inequality $\sqrt{\ln(3+x+y)} \le 2 \sqrt{\ln(3+x)} \sqrt{\ln(3+y)} \quad (x,y>0)$

Question

Let $x>0$ and $y>0$. Prove that: $$\sqrt{\ln(3+x+y)} \le 2 \sqrt{\ln(3+x)} \sqrt{\ln(3+y)}.$$

I'm wondering about a proof approach here. Obviously I can square both sides, but any general suggestions beyond that?

Answer 1

Let $x\geq y$.

Thus, $$4\ln(3+x)\ln(3+y)\geq\ln(3+x)^4>\ln(3+2x)\geq\ln(3+x+y)$$