I want to prove the following inequality :
$$\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab, $$ $a,b,c,d \in \mathbb{R}_{+} .$
In my book, at the answers chapter the author uses AM $\geq$ GM, but I haven't any idea how I can use that.
Thanks :)
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$\begingroup$ The RHS $= abcd(\sum\frac{1}{a})$ $\endgroup$ – lab bhattacharjee Aug 18 '12 at 16:13
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$\begingroup$ The author did't give a complete answer. He suggested that I could resove this inequality using AM $\geq$ GM. $\endgroup$ – Iuli Aug 18 '12 at 20:38
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$\begingroup$ @Iuli: what is the book and the name of the author? $\endgroup$ – user 1591719 Aug 18 '12 at 20:40
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$\begingroup$ The name of the book is : Inequalities-Theorems-Techniques-and-Selected-Problems and the name of the author is Zdravko-Cvetkovski. If you don't have this book I can send you. This inequality you can find in exercise 113, in the last part of the proof. $\endgroup$ – Iuli Aug 18 '12 at 20:43
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$\begingroup$ @Iuli: I know the book and it's worth to have it. $\endgroup$ – user 1591719 Aug 18 '12 at 21:06
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Also notice that:
$$(a+b+c+d)^3 - 16(abc+abd+acd+bcd) = (a+b+c+d)(a+b-c-d)^2 + 4(c-d)^2(a+b) + 4(a-b)^2(c+d) \ge 0$$ Or $$(a+b)[(a+b-c-d)^2 + 4(c-d)^2] + (c+d)[(a+b-c-d)^2 + 4(a-b)^2]\ge0$$
This way is suggested by a friend of mine.
answered Aug 18 '12 at 16:12
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1$\begingroup$ Please, can you give me a more conclusive answer. Thank :) $\endgroup$ – Iuli Aug 18 '12 at 16:19
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$\begingroup$ In the first suggestion, terms are either squared terms, and therefore positive, or involve adding together positive numbers, therefore positive. $\endgroup$ – Kenshin Sep 15 '12 at 8:30
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I posted this inequality on http://www.artofproblemsolving.com/ and I received a nice answer. This answer can be checked on the following link : http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=494463 .
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$\begingroup$ Oh. A brilliant way! It wasn't that hard at all. I went for a while that way, but then I dropped it. I don't remember why. :-) $\endgroup$ – user 1591719 Aug 19 '12 at 10:43
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$\begingroup$ Yes, it is a nice solution. Yesterday I told you about that exercise from that book - now I need to find a solution for that inequality using Cauchy-Schwarz. Can you help me, please? I will post the inquality immediately . Thanks:) $\endgroup$ – Iuli Aug 19 '12 at 11:03
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You need to show that $$(a + b + c + d)^3 - 16(abc + bcd + cda + dab) \geq 0$$ It suffices to show this on any set of the form $0 \leq a,b,c,d \leq N$. By calculus (the "extreme value theorem") the function $(a + b + c + d)^3 - 16(abc + bcd + cda + dab)$ achieves its minimum at some $(a,b,c,d)$ in the set $0 \leq a,b,c,d \leq N$. I claim that this minimum has to occur when $a = b = c = d$.
Write $$(a + b + c + d)^3 - 16(abc + bcd + cda + dab) = (a + b + c + d)^3 - 16(a + b)cd - 16(c + d)ab$$ Note that by AM-GM, we have $16(c + d)ab \leq 16(c + d)({a + b \over 2})^2$. If we had $a \neq b$, we could replace $a$ and $b$ by ${a + b \over 2}$, leaving $c$ and $d$ constant, and we'd get a smaller value. So since $(a,b,c,d)$ is the minimum, this can't happen and we conclude that $a = b$. For similar reasons $c = d$, reversing the roles of the terms $16(a + b)cd$ and $16(c + d)ab$
Next, write $$(a + b + c + d)^3 - 16(abc + bcd + cda + dab) = (a + b + c + d)^3 - 16(b + d)ac - 16 (a + c)bd$$ Then arguing like above gives $b = d$ and $a = c$. Combining the above gives $a = b = c = d$, whereupon $(a + b + c + d)^3 - 16(abc + bcd + cda + dab) = (4a)^3 - 16(4a^3) = 0$. Since this is the minimum, the expression $(a + b + c + d)^3 - 16(abc + bcd + cda + dab)$ is nonnegative as needed.
