Prove that (((((a+b)(b+c)(c+a))/8) ))^(1/3) ≥ (√((ab+bc+ca)/3)) for a,b,c


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Question Number 120215 by benjo_mathlover last updated on 30/Oct/20

$P r o v e t h a t \sqrt[3]{\frac{(a + b) (b + c) (c + a)}{8}} ⩾ \sqrt{\frac{a b + b c + c a}{3}}$ $f o r a, b, c > 0$
	Answered by TANMAY PANACEA last updated on 30/Oct/20

	$\frac{a + b}{2} ⩾ \sqrt{a b}$ $\frac{b + c}{2} ⩾ \sqrt{b c}$ $\frac{c + a}{2} ⩾ \sqrt{a c}$ $m u l t i p l y t h e m$ $\frac{(a + b) (b + c) (c + a)}{8} ⩾ \sqrt{a^{2} b^{2} c^{}}$ $\frac{(a + b) (b + c) (c + a)}{8} ⩾ a b c$ ${[\frac{(a + b) (b + c) (c + a)}{8}]}^{\frac{1}{3}} ⩾ {(a b c)}^{\frac{1}{3}}$ $n o w \sqrt{\frac{a b + b c + a c}{3}} ⩾ \sqrt{{(a b \times b c \times a c)}^{\frac{1}{3}}}$ $\sqrt{\frac{a b + b c + a c}{3}} ⩾ {(a^{2} b^{2} c^{2})}^{\frac{1}{6}}$ $\sqrt{\frac{a b + b c + a c}{3}} ⩾ {(a b c)}^{\frac{1}{3}}$
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