I started to study inequalities - I try to solve a lot of inequlites and read interesting .solutions . I have a good pdf, you can view from here . The inequality which I tried to solve and I didn't manage to find a solution can be found in that pdf but I will write here to be more explicitly.

Exercise 1.3.4(a) Let $a,b,c$ be positive real numbers. Prove that $$\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4.$$

(b) For real numbers $a,b,c \gt0$ and $n \leq3$ prove that: $$\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+n\left(\frac{3\sqrt[3]{abc}}{a+b+c} \right)\geq 3+n.$$

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    $\begingroup$ Please avoid using $$__$$ in the title. $\endgroup$ – Sasha Aug 31 '12 at 5:19

First, I will prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3$, and then I will prove that $\displaystyle\frac{3 \sqrt[3]{abc}}{a+b+c}$.

1) $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3$

Let's call $\varphi(x_1,x_2,x_3)=\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_1}$. Le'ts prove that $\varphi$ is minimized when $x_1 = x_2 = x_3$: $\frac{\partial\varphi}{\partial x_i} = \frac{1}{x_{i+1}} - \frac{x_{i-1}}{x_i^2} = \frac{x_i^2 - x_{i-1}x_{i+1}}{x_i^2 x_{i+1}}$ (with $x_4 = x_1$ and $x_0 = x_3$).

So $\Delta\varphi = 0$ is equivalent to : $$x_1^2 = x_2 x_3$$ $$x_2^2 = x_1 x_3$$ $$x_3^2 = x_1 x_2$$

By dividing one equation by another, we get that $x_1 = x_2 = x_3$. We need to prove that $\varphi$ is convex, to show that this function is minimized when its gradient is $0$. For this, let's compute its Hessian matrix $M$:

$$M = (M_{ij}) = (\frac{\partial^2\varphi}{\partial x_i\partial x_j})$$ We have: $$M_{ii} = \frac{2 x_{i-1}}{x_i^3}$$ $$M_{i(i+1)} = -\frac{1}{x_{i+1}^2}$$ $$M_{i(i-1)} = -\frac{1}{x_i^2}$$

Which gives us:

$$M = \left( \begin{array}{ccc} \frac{2 x_3}{x_1^3} && -\frac{1}{x_2^2} && -\frac{1}{x_1^2}\\ -\frac{1}{x_2^2} && \frac{2 x_1}{x_2^3} && -\frac{1}{x_3^2}\\ -\frac{1}{x_1^2} && -\frac{1}{x_3^2}&& \frac{2 x_2}{x_3^3} \end{array} \right)$$

Calculating its determinant and verifying it is positive will show that $\varphi$ is convex, which achives to prove inequality 1)

2) $\displaystyle\frac{3 \sqrt[3]{abc}}{a+b+c}\geq 1$ This is equivalent to $\sqrt[3]{abc}\geq\frac{a+b+c}{3}$, or $\ln(\frac{a+b+c}{3})\geq \frac{1}{3}(\ln a +\ln b+\ln c)$, which is true because the function $\ln$ is concave.

By combining those two inequalities, we instantly solve the two questions.

  • $\begingroup$ in part 2 your inequality is in the wrong direction. By AM-GM arithmetic average is larger than geometric average. $\endgroup$ – Maesumi Sep 12 '12 at 0:18
  • $\begingroup$ @Maesumi that's right, I made a mistake after taking the log, corrected now. Thanks for pointing it out! $\endgroup$ – S4M Sep 12 '12 at 7:54
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    $\begingroup$ @S4M By AM-GM, $\sqrt[3]{abc}\leq\frac{a+b+c}{3}$. This is a well-known inequality. $\endgroup$ – Arthur Sep 12 '12 at 7:59
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    $\begingroup$ -1 Part 2 is wrong, as pointed out by Arthur. @Iuli Please unaccept this. $\endgroup$ – Calvin Lin Nov 11 '13 at 14:48

Write $$\frac ab+\frac ab+\frac bc\geq \frac{3a}{\sqrt[3]{abc}}$$ by AM-GM.

You get $$\operatorname{LHS} \geq \frac{a+b+c}{\sqrt[3]{abc}}+n\left(\frac{\sqrt[3]{abc}}{a+b+c}\right).$$

Set $$z:=\frac{a+b+c}{\sqrt[3]{abc}}$$ and then notice that for $n\leq 3$, $$z+\frac{3n}{z}\geq 3+n.$$ Indeed the minimum is reached for $z=\sqrt{3n}\leq 3$; since $z\geq 3$, the minimum is reached in fact for $z=3$.


By C-S $$\sum_{cyc}\frac{a}{b}=\sum_{cyc}\frac{a^2}{ab}\geq\frac{(a+b+c)^2}{ab+ac+bc}.$$ Thus, it's enough to prove that $$\frac{(a+b+c)^2}{ab+ac+bc}+\frac{3\sqrt[3]{abc}}{a+b+c}\geq4.$$ Now, let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.

Hence, our inequality it's $f(v^2)\geq0,$ where $f$ decreases, which says that it's enough to prove the last inequality for a maximal value of $v^2$, which happens for equality case of two variables.

Since the last inequality is homogeneous, we can assume $b=c=1$. Also, let $a=x^3$.

Id est, we need to prove that $$\frac{(x^3+2)^2}{2x^3+1}+\frac{3x}{x^3+2}\geq4$$ or $$(x-1)^2(x^6+2x^5+3x^4+2x^3+x^2+6x+3)\geq0.$$ Done!

The following inequality is also true.

Let $a$, $b$ and $c$ be positives. Prove that: $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{24\sqrt[3]{abc}}{a+b+c}\geq11.$$

  • $\begingroup$ what is C-S? Can you explain $\endgroup$ – user500668 Aug 2 '18 at 15:28
  • $\begingroup$ It's Cauchy-Schwarz inequality: For all reals $a_i$ and positives $b_i$ we have: $\frac{a_1^2}{b_1}+\frac{a_2^2}{b_2}+...+\frac{a_n^2}{b_n}\geq\frac{(a_1+a_2+...+a_n)^2}{b_1+b_2+...+b_n}.$ $\endgroup$ – Michael Rozenberg Aug 2 '18 at 16:02
  • $\begingroup$ oh thanks, I didn't know that. $\endgroup$ – user500668 Aug 2 '18 at 16:20

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