# Questions tagged [a.m.-g.m.-inequality]

For questions about proving and manipulating the AM-GM inequality. To be used necessarily with the [inequality] tag.

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### Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
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### New bound for Am-Gm of 2 variables

Today I'm interested by the following problem : Let $x,y>0$ then we have : $$x+y-\sqrt{xy}\leq\exp\Big(\frac{x\ln(x)+y\ln(y)}{x+y}\Big)$$ The equality case comes when $x=y$ My proof uses ...
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### Inequality with five variables

Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+e}+\frac{e}{e+a}\geq\frac{a+b+c+d+e}{a+b+c+d+e-3\sqrt[5]{abcde}}$$ Easy to show ...
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### Tighter inequality than Cauchy - Schwarz inequality

I just learned today that there can be a tighter result than AM-GM (Arithmetic Mean - Geometric Mean) inequality. In particular: Let $a, b > 0$ then \label{1}\tag{1} \dfrac{a+b}{2}...
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### Prove that $a^4+b^4+1\ge a+b$.

Prove that $$a^4+b^4+1\ge a+b$$ for all real numbers $a,b$. What I've tried: 1.I checked how AM-GM may help but doesn't look like it's useful here. I've tried: $$(a^2+b^2)^2 -2(ab)^2+1 \ge a+b$$ ...
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### How to compare logarithms $\log_4 5$ and $\log_5 6$?

I need to compare $\log_4 5$ and $\log_5 6$. I can estimate both numbers like $1.16$ and $1.11$. Then I took smallest fraction $\frac{8}{7}$ which is greater than $1.11$ and smaller than $1.16$ and ...
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