Simple inequality

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Simple inequality

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Source: Daniel Sitaru

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sqing

#1 h Feb 10, 2017, 8:31 AM • 2 Y

Y by Adventure10, Mango247

Let $a,b,c>0$ . Prove that $\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+9>\frac{3}{2}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$

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#2 h Feb 10, 2017, 9:45 AM • 2 Y

Y by Adventure10, Mango247

The following inequality is also true
//cdn.artofproblemsolving.com/images/e/1/f/e1f25cdd5a4f78453076de843e276c26c9d1d1e1.jpg

//cdn.artofproblemsolving.com/images/e/1/f/e1f25cdd5a4f78453076de843e276c26c9d1d1e1.jpg

This post has been edited 1 time. Last edited by luofangxiang, Feb 10, 2017, 9:45 AM

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42088 posts

sqing

#3 h Feb 10, 2017, 12:43 PM • 2 Y

Y by Adventure10, Mango247

luofangxiang wrote:

The following inequality is also true

http://s16.sinaimg.cn/middle/006ptkjAzy78FZvEoXB1f&690

Yeah.
Let $a,b,c>0$ . Prove that $\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+6\geq\frac{3}{2}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$ Equality holds when $a=b=c$

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arqady

#4 h Feb 10, 2017, 8:03 PM • 2 Y

Y by Adventure10, Mango247

Let $a$ , $b$ and $c$ be positive numbers. Prove that:
$\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+\frac{22}{3}\geq\frac{31}{18}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$

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arandomperson123

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arandomperson123

#5 h Feb 10, 2017, 8:18 PM • 1 Y

Y by Adventure10

can someone actually provide a proof?

EDIT: thanks @below

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arqady

#6 h Feb 10, 2017, 8:25 PM • 2 Y

Y by Adventure10, Mango247

arandomperson123, use uvw.

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sqing

#7 h Feb 11, 2017, 12:39 AM • 1 Y

Y by Adventure10

arqady wrote:

Let $a$ , $b$ and $c$ be positive numbers. Prove that:
$\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+\frac{22}{3}\geq\frac{31}{18}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$

Very nice.

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sqing

#8 h Apr 30, 2025, 9:50 AM

Y by

sqing wrote:

Let $a,b,c>0$ . Prove that $\frac{a^3}{b^3}+\frac{b^3}{c^3}+\frac{c^3}{a^3}+9>\frac{3}{2}\left(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}+ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)$

Let $a , b , c$ be positive real numbers such that $abc=1$ . Prove the inequality:
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} +3 \geq 2(\frac{a}{b}+\frac{b}{c}+\frac{c}{a})$ h

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