Let a, b ,c be positive real number such that a+b+c=1 , Prove that:
\sqrt{1+\dfrac{bc }{a}}+\sqrt{1+\dfrac{ca }{b}}+\sqrt{1+\dfrac{ab }{c}}\ge 2\sqrt{3}.
Solution
We will show that
\left(1+\dfrac{bc}{a}\right)\left(1+\dfrac{ca}{b}\right)\left(1+\dfrac{ab}{c}\right) \ge \dfrac{64}{27}
Note that
\prod_{cyc}\left(1+\dfrac{bc}{a}\right) =\prod_{cyc}\dfrac{a(a+b+c)+bc}{a} =\prod_{cyc}\dfrac{(a+b)(a+c)}{a}
=\dfrac{(a+b)^2(b+c)^2(c+a)^2}{abc}
=\dfrac{(a+b)^2(b+c)^2(c+a)^2}{abc}
Again using the well known inequality
(b+c)(c+a)(a+b) \ge \dfrac{8}{9}(a+b+c)(ab+bc+ca)
We get that
\prod_{cyc}\left(1+\dfrac{bc}a\right)
=\dfrac{(a+b)^2(b+c)^2(c+a)^2}{abc} \ge \dfrac{64(a+b+c)^2(ab+bc+ca)^2}{81abc}
\ge \dfrac{64.3abc(a+b+c)}{81abc}=\dfrac{64}{27}
=\dfrac{(a+b)^2(b+c)^2(c+a)^2}{abc} \ge \dfrac{64(a+b+c)^2(ab+bc+ca)^2}{81abc}
\ge \dfrac{64.3abc(a+b+c)}{81abc}=\dfrac{64}{27}
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