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Question: For $a,b,c,d > 0$ and $abcd = 1$, show that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$$

My Attempts: Trivial to see that equality occurs for $a = b= c = d$ - so this indicates to use some inequality wherin the equality condition holds when all quantities are equal - I've tried AM-GM-HM, Power means, Rearrangement, Cauchy-Schwarz, Newton, Maclaurin, Weighted AM-GM - it doesn't help, and the reason is that these inequalities obtain a function of $a,b,c,d$ on the RHS whose minimum is $<7$ - and that minimum and the inequality can never hold simultaneously - giving a loose bound.

I've tried rearranging the inequality to the form $(\sum_{cyc} a )(\sum_{cyc} \frac{1}{a}) - 7 (\sum_{cyc} a) + 12 \geq 0 $ - This doesn't help. I've tried the substitution $(a,b,c,d) \rightarrow (\frac{1}{x}, \frac{1}{y}, \frac{1}{z}, \frac{1}{w} )$ - this is of no help either. Even tried the substitution $(a,b,c,d) \rightarrow (\frac{x}{y}, \frac{y}{z}, \frac{z}{w}, \frac{w}{x} )$ - of no help either. Writing it as $\frac{4}{M(-1)} + \frac{3}{M(1)} \geq 7$ where $M(x)$ is the power means function, and hoping to use some property of $M(x)$ doesn't work either.

My Analysis: Suppose $S_i$ denotes the sum taken $i$ at a time. It is trivial to see that $S_3 = \sum_{cyc} \frac{1}{a}$ is lower bounded by $S_4$ by Maclaurin or GM-HM. However $S_1$ is unbounded above even if $S_4 = 1$. However when $S_1$ reaches a very high value, so does $S_3$, so a minimum has to involve considering both terms simultaneously i.e. you can't minimize each separately and then hope to minimize the function.

Brute Force Solution: The brute force solution of considering it as a function of 4 variables and then optimizing obviously works, I'm hoping for something more smarter or elegant.

Michael Rozenberg
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Anon
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2 Answers2

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Mixing Variables and $uvw$ help.

I'll post my solution, which I found eight years ago.

Let $f(a,b,c,d)=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{12}{a+b+c+d}- 7$ and $a=\max\{a,b,c,d\}$.

Thus, $$f(a,b,c,d)-f\left(a,\sqrt[3]{bcd},\sqrt[3]{bcd},\sqrt[3]{bcd}\right)=$$ $$=\frac{bc+bd+cd-3\sqrt[3]{b^2c^2d^2}}{bcd}-\frac{12\left(b+c+d-3\sqrt[3]{bcd}\right)}{(a+b+c+d)\left(a+3\sqrt[3]{bcd}\right)}\geq$$ $$\geq\frac{bc+bd+cd-3\sqrt[3]{b^2c^2d^2}}{bcd}-\frac{12\left(b+c+d-3\sqrt[3]{bcd}\right)}{(\frac{b+c+d}{3}+b+c+d)\left(\frac{b+c+d}{3}+3\sqrt[3]{bcd}\right)}=$$ $$=\frac{bc+bd+cd-3\sqrt[3]{b^2c^2d^2}}{bcd}-\frac{27\left(b+c+d-3\sqrt[3]{bcd}\right)}{(b+c+d)\left(b+c+d+9\sqrt[3]{bcd}\right)}.$$ We'll prove that $$\frac{bc+bd+cd-3\sqrt[3]{b^2c^2d^2}}{bcd}-\frac{27\left(b+c+d-3\sqrt[3]{bcd}\right)}{(b+c+d)\left(b+c+d+9\sqrt[3]{bcd}\right)}\geq0.$$ Indeed, let $b+c+d=3u$, $bc+bd+cd=3v^2$ and $bcd=w^3$.

Thus, $$\frac{bc+bd+cd-3\sqrt[3]{b^2c^2d^2}}{bcd}-\frac{27\left(b+c+d-3\sqrt[3]{bcd}\right)}{(b+c+d)\left(b+c+d+9\sqrt[3]{bcd}\right)}\geq0$$ it's $ g(v^2)\geq0,$ where $g$ is a linear increasing function.

Id est, $g$ gets a minimal value, when $v^2$ gets a minimal value, which happens for equality case of two variables.

Since $g(v^2)\geq0$ is homogeneous inequality, it's enough to check one case only: $c=d=1$, which after substitution $b=x^3$ gives $$(x-1)^2(2x^7+x^6+18x^5-10x^4-50x^3+36x^2+26x+4)\geq0,$$ which is true.

Id est, $$f(a,b,c,d)\geq f\left(a,\sqrt[3]{bcd},\sqrt[3]{bcd},\sqrt[3]{bcd}\right)$$ and it's enough to prove that $f(a,b,b,b)\geq0$, where $a=\frac{1}{b^3}$, which gives $$(b-1)^2(3b^6+6b^5+9b^4-9b^3-5b^2-b+3)\geq0,$$ which is true.

Michael Rozenberg
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  • I'll take me a while to go through this - but could you say give some insight as to how you formed your approach - was it a lucky idea or is there some theory behind it (cause I'm still a beginner, I wouldn't know) - i.e what ideas could I take back from this question? No offence it's a great method, but tbh optimizing over 4 variables would be more concise. – Anon Jun 05 '19 at 18:37
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    @Kaind It was optimizing over 4 variables. There are very many methods of inequalities proofs, but non of those methods are working in the general. In our case the Mixing Variables method helps. – Michael Rozenberg Jun 05 '19 at 18:43
  • There's something called a 'mixing variables method' - could you point me to some link or pdf which explains this? (By optimizing 4 variables, I meant finding saddle points and critical points, and then getting answer). – Anon Jun 05 '19 at 18:45
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    For those questions visit the AOPS website. – Dr. Sonnhard Graubner Jun 05 '19 at 18:52
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    @Kaind About MV you can read here: https://math.stackexchange.com/questions/tagged/mixing-variables Now I see that Lagrange multipliers helps, but for the proof we need to consider two cases: 1. $b=c=a$; 2. $b=a$ and $c=d$. – Michael Rozenberg Jun 05 '19 at 18:52
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Here is a proof via Sum of Squares (SOS).

We have \begin{align*} &\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + abc + \frac{12}{a + b + c + 1/(abc)} - 7\right)abc(a^2bc + ab^2c + abc^2 + 1)\\ ={}& \sum_{\mathrm{cyc}} \frac{abc^2 (10ab - 7a - 7b + 4)^2}{60} + \sum_{\mathrm{cyc}}\frac{ab}{144}(12abc^2 - 5ac - 5bc - 14c + 12)^2\\ &\qquad + \sum_{\mathrm{cyc}} \frac{7abc^2(a + b -2)^2}{720}. \end{align*}

River Li
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