Suppose a,b,ca,b,ca,b,c be 3 positive reals such that
a+b+c≥ab+bc+caa+b+c≥\frac{a}{b}+\frac{b}{c}+\frac{c}{a}a+b+c≥ba+cb+ac
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}b(c+a)a3c+c(a+b)b3a+a(b+c)c3b≥23.
I cannot still solve this problem.Please,someone help me .
Note by Souryajit Roy 6 years, 11 months ago
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2 \times 3
2^{34}
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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)2(a+b+c)(ba+cb+ac)(cyc∑b(c+a)a3c)
≥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) ≥Ho¨lder(a+b+c)3≥?3(a+b+c)(ba+cb+ac)
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)^23(ba+cb+ac)≤3(a+b+c)≤?(a+b+c)2
a+b+c≥ab+bc+ca≥AM-GM3. □\displaystyle a+b+c\ge \frac{a}{b} +\frac{b}{c} +\frac{c}{a}\stackrel{\text{AM-GM}}\ge 3.\:\:\squarea+b+c≥ba+cb+ac≥AM-GM3.□
i think you should try to use AM-GM inequality
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I have tried...but it was of no use
search up holder's inequality
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to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
2(a+b+c)(ba+cb+ac)(cyc∑b(c+a)a3c)
≥Ho¨lder(a+b+c)3≥?3(a+b+c)(ba+cb+ac)
3(ba+cb+ac)≤3(a+b+c)≤?(a+b+c)2
a+b+c≥ba+cb+ac≥AM-GM3.□
i think you should try to use AM-GM inequality
Log in to reply
I have tried...but it was of no use
search up holder's inequality