Inequality proof by using the Muirhead inequality.

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### Problem in normalization inequalities

Let $a,b,c>0$. Prove that: $$\dfrac{(a+b)^2(b+c)^2(c+a)^2}{abc} \ge \dfrac{64}{27}(a+b+c)^3$$ First solution: $\bullet$ Since the inequality is homogeneous, we may normalize $a+b+c=3$, we need to ... 135 views

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### Question on the Proof of Muirhead's Inquality (cited from AoPS) [closed]

In AoPS' (Art of Problem Solving) proof of Muirhead's inequality, how does the below equality work out? The below equation appears to show two expressions (1 and 2), each under the symmetric sum ...
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Let a, b, c be positive real numbers. Prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+ \frac{3\cdot\sqrt{abc}}{a+b+c} \geq 4$ Ohk now i know using AM-GM that $\frac{a}{b}+\frac{b}{c}+\frac{c}{... -1 votes 3 answers 112 views ### Inequality question . [duplicate] Let$a,b,c>0$with$ \frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$. Prove that$(a + 1)(b + 1)(c + 1) \geq 64$Ohk so we are given that$abc=a+b+c$with that now the inequality becomes$2abc+(a+b+c)+1 \...
Let $a,b,c>0$ with $a+b+c=1$. Show that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq 3 + 2\cdot\frac{\left(a^3 + b^3 + c^3\right)}{abc}$$ Ohhhkk. So first off, \begin{align} a^3 + b^3+ c^3 &...