The inequality presented in the image is a form of Harmonic Mean vs Geometric Mean Inequality, commonly written as:

$\frac{n}{\sum_{k=1}^{n} \frac{1}{a_k}} \leq \sqrt[n]{\prod_{k=1}^{n} a_k}$

The goal is to prove that the harmonic mean of a set of positive numbers $a_1, a_2, \dots, a_n$ is always less than or equal to their geometric mean.

Steps for Proof:

1. Harmonic Mean Definition: The harmonic mean of $n$ positive numbers $a_1, a_2, \dots, a_n$ is given by: $H = \frac{n}{\sum_{k=1}^{n} \frac{1}{a_k}}$ This is the left-hand side of the inequality.

2. Geometric Mean Definition: The geometric mean of these numbers is defined as: $G = \sqrt[n]{\prod_{k=1}^{n} a_k}$ This is the right-hand side of the inequality.

3. Arithmetic Mean-Geometric Mean (AM-GM) Inequality: The AM-GM Inequality states that for any set of positive numbers $x_1, x_2, \dots, x_n$: $\frac{x_1 + x_2 + \dots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \dots x_n}$ This inequality helps us compare the arithmetic mean with the geometric mean, but we are concerned with the harmonic mean, which has a similar relationship.

4. Jensen's Inequality: The inequality we are dealing with can also be seen as a special case of Jensen's Inequality, which applies to convex functions. Since the harmonic mean can be derived using the convexity of the function $f(x) = 1/x$, Jensen's inequality provides the desired result.

Step-by-Step Proof:

• Consider the harmonic mean: $\frac{n}{\sum_{k=1}^{n} \frac{1}{a_k}} \leq \sqrt[n]{\prod_{k=1}^{n} a_k}$ To prove this, apply the AM-GM inequality to the numbers $1/a_1, 1/a_2, \dots, 1/a_n$:

  By AM-GM inequality: [ \frac{1/a_1 + 1/a_2 + \dots + 1/a_n}{n} \geq \sqrt[n]{\left( \frac{1}{a_1} \cdot \frac{1}{a_2} \dots \cdot \frac{1}{a_n} \right)} = \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} ]

• Taking the reciprocal of both sides: $\frac{n}{1/a_1 + 1/a_2 + \dots + 1/a_n} \leq \sqrt[n]{a_1 a_2 \dots a_n}$

This completes the proof of the inequality, showing that the harmonic mean is less than or equal to the geometric mean.

Would you like more details on any of the steps or clarifications?

Here are 5 related questions for further exploration:

1. How is the arithmetic mean related to the geometric and harmonic means?
2. Can you apply the AM-GM inequality to non-positive numbers?
3. What is Jensen’s inequality, and how is it used in proofs like this?
4. What are some real-life applications of harmonic and geometric means?
5. How does this inequality extend to weighted harmonic and geometric means?

Tip: Always remember that inequalities like AM-GM-HM help in comparing different types of averages, with geometric means often playing a key role in multiplicative contexts.