We are asked to prove the inequality involving the Arithmetic Mean (AM), Harmonic Mean (HM), and Geometric Mean (GM) for two values $X_1$ and $X_2$. Specifically, we need to prove that:

$\text{AM} \times \text{HM} > (\text{GM})^2$

Definitions:

Let $X_1$ and $X_2$ be two positive real numbers. The definitions of the means are as follows:

• Arithmetic Mean (AM): $\text{AM} = \frac{X_1 + X_2}{2}$
• Harmonic Mean (HM): $\text{HM} = \frac{2X_1 X_2}{X_1 + X_2}$
• Geometric Mean (GM): $\text{GM} = \sqrt{X_1 X_2}$

Proof:

We need to prove: $\text{AM} \times \text{HM} > (\text{GM})^2$ Substituting the expressions for AM, HM, and GM into the inequality:

$\left( \frac{X_1 + X_2}{2} \right) \times \left( \frac{2X_1 X_2}{X_1 + X_2} \right) > \left( \sqrt{X_1 X_2} \right)^2$

Simplifying the left-hand side: $\left( \frac{X_1 + X_2}{2} \right) \times \left( \frac{2X_1 X_2}{X_1 + X_2} \right) = X_1 X_2$

Now, the inequality becomes: $X_1 X_2 > X_1 X_2$

This is clearly not true in general; it holds as an equality, meaning this attempt fails to prove the original statement.

Alternative Perspective:

The correct inequality relation between the means is: $\text{AM} \geq \text{GM} \geq \text{HM}$

Thus, for the given case, the true relation between the means is: $\text{AM} \times \text{HM} = (\text{GM})^2$

So, the original inequality is incorrect. Instead, $\text{AM} \times \text{HM}$ equals $(\text{GM})^2$.

Would you like further details or another exploration of this? Here are related questions to deepen your understanding:

1. What are the implications of the inequality $\text{AM} \geq \text{GM} \geq \text{HM}$ for more than two numbers?
2. How can we generalize the proof for any set of $n$ numbers?
3. Why does equality hold between $\text{AM} \times \text{HM}$ and $(\text{GM})^2$ in this specific case?
4. What is the significance of these mean inequalities in statistics?
5. How does the proof change if the values $X_1$ and $X_2$ are not positive?

Tip: The inequality relations between different means are widely used in optimization problems and inequalities like the Cauchy-Schwarz inequality.