<s>[INST] Prove for all $a, b, c \in \mathbb {R^+}$ ${a\over\sqrt{a+b}} + {b\over\sqrt{b+c}} + {c\over\sqrt{c+a}} \gt \sqrt{a+b+c}$ is true 
Prove for all $a, b, c \in \mathbb {R^+}$ $${a\over\sqrt{a+b}} + {b\over\sqrt{b+c}} + {c\over\sqrt{c+a}} \gt \sqrt{a+b+c}$$ is true

I have been studying about inequalities and how to prove different inequalities, and I stumbled over this question which I failed to answer. I failed to see a connection between $\sqrt{a+b+c} \text{ and} {a\over\sqrt{a+b}} + {b\over\sqrt{b+c}} + {c\over\sqrt{c+a}}$. Is there any way I can prove this question and if there is can you please help. Thank you.
 [/INST] You need only Jensen applied to convex function $f(x)={1 \over \sqrt{x}}$:
$$LHS=a\space f(a+b)+b\space f(b+c)+c\space f(c+a)\ge(a+b+c)\space f(\frac{a(a+b)+b(b+c)+c(c+a)}{a+b+c})$$
$$LHS\ge(a+b+c)\space f(\frac{(a+b+c)^2-ab-bc-ac}{a+b+c})$$
$$LHS\ge(a+b+c)\sqrt{\frac{a+b+c}{(a+b+c)^2-ab-bc-ac}}$$
$$LHS\gt(a+b+c)\sqrt{\frac{a+b+c}{(a+b+c)^2}}$$
$$LHS\gt\sqrt{a+b+c}$$
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