Inequality Problem

Suppose a,b,ca,b,c be 3 positive reals such that

a+b+cab+bc+caa+b+c≥\frac{a}{b}+\frac{b}{c}+\frac{c}{a}

Show that a3cb(c+a)+b3ac(a+b)+c3ba(b+c)32\frac{a^{3}c}{b(c+a)}+\frac{b^{3}a}{c(a+b)}+\frac{c^{3}b}{a(b+c)}≥\frac{3}{2}.

I cannot still solve this problem.Please,someone help me .

#Algebra #CauchySchwarzInequality #Inequality

Note by Souryajit Roy
6 years, 11 months ago

No vote yet
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Comments

2(a+b+c)(ab+bc+ca)(cyca3cb(c+a))\displaystyle 2(a+b+c)\left( \frac{a}{b} +\frac{b}{c} +\frac{c}{a} \right) \left( \sum_{\text{cyc}}\frac{a^3c}{b(c+a)} \right)

Ho¨lder(a+b+c)3?3(a+b+c)(ab+bc+ca)\displaystyle \stackrel{\text{Hölder}}\ge (a+b+c)^3\stackrel{\text{?}}\ge 3(a+b+c)\left( \frac{a}{b} +\frac{b}{c} +\frac{c}{a} \right)

3(ab+bc+ca)3(a+b+c)?(a+b+c)2\displaystyle 3\left( \frac{a}{b} +\frac{b}{c} +\frac{c}{a} \right)\le 3(a+b+c)\stackrel{\text{?}}\le (a+b+c)^2

a+b+cab+bc+caAM-GM3.\displaystyle a+b+c\ge \frac{a}{b} +\frac{b}{c} +\frac{c}{a}\stackrel{\text{AM-GM}}\ge 3.\:\:\square

mathh mathh - 6 years, 10 months ago

i think you should try to use AM-GM inequality

akash deep - 6 years, 10 months ago

Log in to reply

I have tried...but it was of no use

Souryajit Roy - 6 years, 10 months ago

search up holder's inequality

blah blah - 6 years, 10 months ago
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