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Solving Inequality Problem

Let $$x, y, z$$ be positive real numbers . Prove that:

$\displaystyle \frac{x^{y+z}}{(y+z)^2}+\frac{y^{z+x}}{(z+x)^2}+\frac{z^{x+y}}{(x+y)^2}\geq \frac{3}{4}$

I always confused how can I solve inequality with variable-power on it. Appreciate your help to solve this problem for sure :)

Note by Natasha Astriani
3 years, 5 months ago

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I thought of this on my own but I saw that Cody Johnson has already shared it.

So, showing that $$f$$ is convex for the interval [0,S] ( Take second-order derivative or the 'definition' method)

Applying Jensen's inequality for $$f(a) = \frac{{a}^{S-a}}{{S-a}^{2}}$$,

$$f(x) + f(y) + f(z) \geq 3\frac{x+y+z}{3}$$

$$\geq 3\frac{S}{3}$$

$$\geq 3\frac{{(\frac{S}{3})}^{\frac{2S}{3}}}{{\frac{2S}{3}}^{2}}$$

$$\geq 3\frac{{(\frac{S}{3})}^{\frac{2S-6}{3}}}{4}$$ --------- 1

Now, we need to find the minimum value of $${(\frac{S}{3})}^{\frac{2S-6}{3}}$$. Taking derivative of it w.r.t. S and equating to 0,

$${(\frac{S}{3})}^{\frac{2S-6}{3}}(\frac{2S-6}{3S} + \frac{2}{3}ln(\frac{S}{3})) = 0$$ [ln(b) is the logarithm base e of b]

So, either $${(\frac{S}{3})}^{\frac{2S-6}{3}} = 0$$, which is not possible as $$S > 0$$

or $$\frac{2S-6}{3S} + \frac{2}{3}ln(\frac{S}{3}) = 0$$

After some bashing, this can be written as,

$$S(ln(\frac{S}{3})) = 3 - S \Rightarrow {e}^{c} = \frac{e}{c}$$ [for c = $$\frac{3}{S}$$], which is only possible for c = 1.

Hence, $$S = 3$$.

Substituting in 1,

$$f(x) + f(y) + f(z) \geq 3\frac{{(\frac{3}{3})}^{\frac{2*3-6}{3}}}{4}$$

which is just

$$f(x) + f(y) + f(z) \geq \frac{3}{4}$$

- 3 years ago

nice solution @Kartik Sharma

- 3 years ago

@Cody Johnson I hope I didn't copy and if I did, then I really didn't mean it!

- 3 years ago

Wow, this is so obvious now that I read your solution... :(

- 3 years ago

- 3 years ago

Let $$s=x+y+z$$ and $$f(x)=\frac{x^{s-x}}{(s-x)^2}$$. Confirm that $$f$$ is convex on $$0<x<s$$. Apply Jensen's inequality.

I'm not sure if this is correct, but I'm pretty darn sure… can't confirm because I'm on my phone

- 3 years ago

We know that $$AM \ge GM$$ and they are equal when $$x=y=z=1$$. Therefore,

$\dfrac {\dfrac {x^{y+z}}{(y+z)^2} + \dfrac {y^{z+x}}{(z+x)^2} + \dfrac {z^{x+y}}{(x+y)^2} } {3} \ge \sqrt [3] {\left( \dfrac {x^{y+z}} {(y+z)^2} \right) \left( \dfrac {y^{z+x}}{(z+x)^2} \right) \left( \dfrac {z^{x+y}}{(x+y)^2} \right) }$

$\dfrac {x^{y+z}}{(y+z)^2} + \dfrac {y^{z+x}}{(z+x)^2} + \dfrac {z^{x+y}}{(x+y)^2} \ge 3 \sqrt [3] {\dfrac {x^{y+z}y^{z+x} z^{x+y}} {(x+y)^2(y+z)^2(z+x)^2} }$

$\dfrac {x^{y+z}}{(y+z)^2} + \dfrac {y^{z+x}}{(z+x)^2} + \dfrac {z^{x+y}}{(x+y)^2} \ge \dfrac {3x^{\frac {1}{3} (y+z)}y^{\frac {1}{3} (z+x)} z^{\frac {1}{3} (x+y)}} {[(x+y)(y+z)(z+x)]^\frac{2}{3}}$

Equality occurs when $$x=y=z=1$$:

$\Rightarrow \dfrac {x^{y+z}}{(y+z)^2} + \dfrac {y^{z+x}}{(z+x)^2} + \dfrac {z^{x+y}}{(x+y)^2} \ge \dfrac {3\dot{} 1\dot{}1\dot{}1} {[2\dot{}2\dot{}2]^\frac{2}{3}} = \dfrac {3}{4}$

- 3 years ago

I don't see how you proved the inequality... You just manipulated it a bit and then claimed that equality happens at $$x=y=z=1$$.

@Mardokay Mosazghi I'll try to solve it. I don't usually tackle problems with variable exponents like this one though.

EDIT: I honestly have no idea how to prove this. Even when assuming $$x=y=z$$, it's pretty non-trivial to prove that the minimum is when $$x=y=z=1$$. I'm guessing that this inequality cannot be proved using elementary techniques.

- 3 years ago

Uhm Let a = x + z, b = y + z, and c = x + y?

- 3 years ago

That's a good idea, to substitute something in. However your substitution (called Ravi's substitution) is only applicable if x, y, and z are the sides of a triangle.

- 3 years ago

Daniel, you are right. I thought I was using the theorem. Can anyone help?

- 3 years ago

Yes its easy u put all x.y.z 1 to prove equality and gv counter example by taking all zero which is not a poitive integer. Hence inequality is not proved on other then positive.. And as it is a polynomial it will form a ring hence close under both operation so inequality is proved..

- 3 years ago

Put x =1 and then 0 it means limit are satisfying. Rest I don't know

- 3 years ago

why do you do this for x only, please elaborate.

- 3 years ago

Check now

- 3 years ago

@Daniel Liu Can you help us with this problem?

- 3 years ago