# Inequality - solve in at least two ways

• MHB
• lfdahl
In summary, inequality is caused by unequal distribution of wealth and resources, discrimination, and biased policies. It can be addressed through government policies and promoting equal opportunities. Inequality can have negative impacts on society, perpetuating cycles of poverty and hindering social mobility. Education has been shown to play a crucial role in reducing inequality. Inequality also has negative effects on health, with individuals from lower socioeconomic backgrounds having limited access to healthcare, healthy food options, and safe living conditions.
lfdahl
Gold Member
MHB
Find in at least two different ways the smallest $\alpha$, such that

$\sqrt[3]{x}+\sqrt[3]{y} \leq \alpha \sqrt[3]{x+y}$

- for all $x,y \in \mathbb{R}_+$

I’ve found one way; I’ll think of another.

We note that $\sqrt[3]8+\sqrt[3]8=4=\sqrt[3]{64}>\sqrt[3]{8+8}$ and so $\alpha\ne1$. Indeed it is clear that we need $\alpha>1$.

Write $X=\sqrt[3]x,\,Y=\sqrt[3]y$. Then we want
$$(X+Y)^3\le\alpha^3(X^3+Y^3)=\alpha^3(X+Y)(X^2-XY+Y^2)$$

$\implies\ (X+Y)^2\le\alpha^3(X^2-XY+Y^2)$

$\implies\ 0\le(\alpha^3-1)X^2+(\alpha^3-2)XY+(\alpha^3-1)Y^2$

$\implies\ 0\le X^2+\dfrac{\alpha^3-2}{\alpha^3-1}XY+Y^2\le\left(\dfrac{2-\alpha^3}{1-\alpha^3}+\dfrac12\right)(X^2+Y^2)$ by AM–GM.

Since $X^2+Y^2\ge0$, we want $\dfrac{\alpha^3-2}{\alpha^3-1}+\dfrac12\ge0$. Hence
$$2(\alpha^3-2)+\alpha^3-1\ge0$$

$\implies\ \alpha\ \ge\ \sqrt[3]{\dfrac53}$.

Olinguito said:
I’ve found one way; I’ll think of another.

We note that $\sqrt[3]8+\sqrt[3]8=4=\sqrt[3]{64}>\sqrt[3]{8+8}$ and so $\alpha\ne1$. Indeed it is clear that we need $\alpha>1$.

Write $X=\sqrt[3]x,\,Y=\sqrt[3]y$. Then we want
$$(X+Y)^3\le\alpha^3(X^3+Y^3)=\alpha^3(X+Y)(X^2-XY+Y^2)$$

$\implies\ (X+Y)^2\le\alpha^3(X^2-XY+Y^2)$

$\implies\ 0\le(\alpha^3-1)X^2+(\alpha^3-2)XY+(\alpha^3-1)Y^2$

$\implies\ 0\le X^2+\dfrac{\alpha^3-2}{\alpha^3-1}XY+Y^2\le\left(\dfrac{2-\alpha^3}{1-\alpha^3}+\dfrac12\right)(X^2+Y^2)$ by AM–GM.

Since $X^2+Y^2\ge0$, we want $\dfrac{\alpha^3-2}{\alpha^3-1}+\dfrac12\ge0$. Hence
$$2(\alpha^3-2)+\alpha^3-1\ge0$$

$\implies\ \alpha\ \ge\ \sqrt[3]{\dfrac53}$.

Hi, Olinguito!

I think, there is a small error in the third line, - should be:

$$0 \leq (\alpha^3-1)X^2-(\alpha^3+2)XY + (\alpha^3-1)Y^2$$

Thanks, Ifdahl.

Yes, there was an error – so I haven’t found anything after all. (Sadface) Oh well, will keep trying.

Hint:

Jensens inequality or power mean inequality or rearrangement inequality

With that hint...

According to the power inequality we have:
$$\frac{X+Y}{2} \le \sqrt[3]{\frac{X^3+Y^3}{2}}$$
with equality iff $X=Y$.
Let $x=X^3$ and $y=Y^3$.
Then:
$$\frac{\sqrt[3]x+\sqrt[3]y}{2} \le \sqrt[3]{\frac{x+y}{2}} \quad\Rightarrow\quad \sqrt[3]x+\sqrt[3]y \le \sqrt[3]4\sqrt[3]{x+y}$$
Thus:
$$\alpha=\sqrt[3]4$$

Jensen’s inequality! Why didn’t I think of that? Stupid me. (Fubar)

The function $f:\mathbb R^+\to\mathbb R^+; t\mapsto\sqrt[3]t$ is concave, so by Jensen’s inequality for concave functions:

$$\frac{\sqrt[3]x+\sqrt[3]y}2\ \le\ \sqrt[3]{\frac{x+y}2}$$

$\implies\ \sqrt[3]x+\sqrt[3]y\ \le\ 2^{\frac23}\sqrt[3]{x+y}.$

Equality is attained when $x=y$; hence $\alpha=2^{\frac23}$.

Klaas van Aarsen said:
With that hint...

According to the power inequality we have:
$$\frac{X+Y}{2} \le \sqrt[3]{\frac{X^3+Y^3}{2}}$$
with equality iff $X=Y$.
Let $x=X^3$ and $y=Y^3$.
Then:
$$\frac{\sqrt[3]x+\sqrt[3]y}{2} \le \sqrt[3]{\frac{x+y}{2}} \quad\Rightarrow\quad \sqrt[3]x+\sqrt[3]y \le \sqrt[3]4\sqrt[3]{x+y}$$
Thus:
$$\alpha=\sqrt[3]4$$

Thankyou, Klaas van Aarsen for your participation and the neat solution, which of course is correct!(Yes)

Olinguito said:
Jensen’s inequality! Why didn’t I think of that? Stupid me. (Fubar)

The function $f:\mathbb R^+\to\mathbb R^+; t\mapsto\sqrt[3]t$ is concave, so by Jensen’s inequality for concave functions:

$$\frac{\sqrt[3]x+\sqrt[3]y}2\ \le\ \sqrt[3]{\frac{x+y}2}$$

$\implies\ \sqrt[3]x+\sqrt[3]y\ \le\ 2^{\frac23}\sqrt[3]{x+y}.$

Equality is attained when $x=y$; hence $\alpha=2^{\frac23}$.

Thankyou, Olinguito! for your participation. Your solution is of course also correct. I also want to thank you for your nice attempt earlier. I have tried to elaborate on your solution path:

I want to thank Olinguito, for the following solution.

Let $a = \sqrt[3]{x}, \: \: b = \sqrt[3]{y}$:$\left ( a+b\right )^3\leq \alpha ^3\left ( a^3+b^3 \right )=\alpha ^3(a+b)\left ( a^2-ab+b^2 \right ) \\\\ \left ( a+b\right )^2\leq \alpha ^3\left ( a^2-ab+b^2 \right )\\\\ (\alpha ^3-1)a^2+(\alpha ^3-1)b^2-(\alpha ^3+2)ab \geq 0 \\\\ a^2+b^2-\frac{\alpha ^3+2}{\alpha ^3-1}ab \geq 0$
Hence, we see, that the fraction: $\frac{\alpha ^3+2}{\alpha ^3-1} \leq 2$

or $\alpha^3 \geq 4$, or $\alpha \geq \sqrt[3]{4}$.

## 1. What are the main causes of inequality?

There are several factors that contribute to inequality, including unequal distribution of wealth and resources, discrimination based on race, gender, or socioeconomic status, and policies that favor certain groups over others.

## 2. How can inequality be solved?

One way to address inequality is through government policies such as progressive taxation and social welfare programs that aim to redistribute wealth and resources. Another approach is through promoting equal opportunities and addressing systemic discrimination.

## 3. How does inequality impact society?

Inequality can have negative effects on society, including increased crime rates, lower levels of social trust and cohesion, and limited economic growth. It can also perpetuate cycles of poverty and hinder social mobility.

## 4. Can education help reduce inequality?

Education has been shown to play a crucial role in reducing inequality. By providing equal access to quality education, individuals from disadvantaged backgrounds have the opportunity to improve their socioeconomic status and break the cycle of poverty.

## 5. How does inequality affect health?

Inequality has been linked to negative health outcomes, as individuals from lower socioeconomic backgrounds may have limited access to healthcare, healthy food options, and safe living conditions. This can lead to higher rates of chronic diseases and decreased life expectancy.

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