If $$a,b,c \in R^+$$, prove $$a^ab^bc^c\geq (abc) iny^{\dfrac{a+b+c}{3} } $$

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If $$a,b,c \in R^+$$, prove $$a^ab^bc^c\geq (abc)	iny^{\dfrac{a+b+c}{3}  } $$

Uniteasy Admin · Accepted Answer

Let $$a\geq b\geq c $$
$$\dfrac{a^ab^bc^c}{ (abc)	iny^{\dfrac{a+b+c}{3} } } = \dfrac{a^{3a}b^{3b}c^{3c}}{(abc)^{a+b+c}}  $$
$$=\dfrac{a^{2a}b^{2b}c^{2c}}{a^{b+c}b^{a+c}c^{a+b}}$$
$$ = \Big( \dfrac{a}{b} \Big) ^{a-b}\Big( \dfrac{b}{c} \Big) ^{b-c}\Big( \dfrac{a}{c} \Big) ^{a-c}$$
$$\geq 1 $$

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