How to prove that $a + 2 \sqrt{a^{2} + b^{2}} + 3 \sqrt{1+a^{2}} \geq 2 + \sqrt{1+b^{2}}$ for all $a,b >0$?

Question

The inequality

$$a + 2 \sqrt{a^{2} + b^{2}} + 3 \sqrt{1+a^{2}} \geq 2 + \sqrt{1+b^{2}}$$

is related to some computation regarding the Shapley value of a stochastic cooperative game. Though I'm not sure it holds, I suspect it is true for $a, b \in \mathbb{R}_{>0}$.

Do you know how to prove this inequality for $a, b \in \mathbb{R}_{>0}$? Or do you have any hints pointing towards a possible proof?

Answer 1

Because $$a + 2 \sqrt{a^{2} + b^{2}} + 3 \sqrt{1+a^{2}} \geq 2b+3\geq b+3\geq2 + \sqrt{1+b^{2}},$$ where the last inequality it's $$b+1\geq\sqrt{1+b^2}$$ or $$2b\geq0,$$ which is obvious.