For positive reals $a,b,c$, prove or disprove $ab+bc+ca+abc\ge \sqrt{ab}+\sqrt{bc}+\sqrt{ca}+1$

Question

For positive reals $a,b,c$, prove or disprove $ab+bc+ca+abc\ge \sqrt{ab}+\sqrt{bc}+\sqrt{ca}+1$

This is an intermediate step of another problem. Its not comming to my head. I have applied AM-GM and CS in all ways my brain can think of.

Answer 1

Let's assume $$a=b=c \Rightarrow 3a^2+a^3\geq 3a+1$$ Now take $a=\frac{1}{3}$ then $$3\cdot\frac{1}{9}+\frac{1}{27}\geq 3\cdot\frac{1}{3}+1$$ which is not true, thus a counter example.

Answer 2

Well, what happens to both sides if $a,b,c$ all become very small?

Answer 3

I think it should read $ab+bc+ca+abc\ge \sqrt{ab}+\sqrt{bc}+\sqrt{ca}+1$.

Try $a=b=c=1/2$. Conclusion ?