Inequalities of Means: AM, GM, QM, HM

Introduction

The inequalities of means are fundamental in mathematics and provide powerful tools for comparing different types of averages. These inequalities establish relationships between the arithmetic mean (AM), geometric mean (GM), quadratic mean (QM), and harmonic mean (HM) of a set of positive real numbers.

The general relationship between these means is expressed as follows:

\[ H_n \leq G_n \leq A_n \leq Q_n \]

where \( H_n \), \( G_n \), \( A_n \), and \( Q_n \) are the harmonic, geometric, arithmetic, and quadratic means, respectively. Equality holds if and only if all the numbers in the set are equal.

Definitions

Given a set of positive real numbers \( a_1, a_2, \dots, a_n \), the means are defined as follows:

• Arithmetic Mean (AM): \[ A_n = \frac{a_1 + a_2 + \dots + a_n}{n} \] The arithmetic mean is the simple average of the numbers.
• Geometric Mean (GM): \[ G_n = \sqrt[n]{a_1 \cdot a_2 \cdot \dots \cdot a_n} \] The geometric mean is the \(n\)th root of the product of the numbers.
• Quadratic Mean (QM): \[ Q_n = \sqrt{\frac{a_1^2 + a_2^2 + \dots + a_n^2}{n}} \] The quadratic mean, also known as the root mean square, is the square root of the average of the squares of the numbers.
• Harmonic Mean (HM): \[ H_n = \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}} \] The harmonic mean is the reciprocal of the average of the reciprocals of the numbers.

Relationship Between the Means

The inequalities among these means can be summarized as follows:

\[ H_n \leq G_n \leq A_n \leq Q_n \]

These inequalities hold for any set of positive real numbers. The proofs of these inequalities can be derived using various methods, including algebraic manipulations, the Cauchy-Schwarz inequality, and Jensen's inequality.

Proof of AM-GM Inequality

The AM-GM inequality states that for any set of positive real numbers \( a_1, a_2, \dots, a_n \), we have:

\[ A_n \geq G_n \]

That is, the arithmetic mean is always greater than or equal to the geometric mean. The equality holds if and only if all the numbers are equal.

Proof for \( n = 2 \) (Simple Case)

Consider two positive real numbers \( a \) and \( b \). The AM-GM inequality states:

\[ \frac{a + b}{2} \geq \sqrt{ab} \]

Squaring both sides:

\[ \left(\frac{a + b}{2}\right)^2 \geq ab \]

Expanding and simplifying:

\[ \frac{a^2 + 2ab + b^2}{4} \geq ab \implies a^2 - 2ab + b^2 \geq 0 \implies (a - b)^2 \geq 0 \]

Since \( (a - b)^2 \) is always non-negative, the inequality holds.

Proof of QM-AM Inequality

The QM-AM inequality states that the quadratic mean is always greater than or equal to the arithmetic mean:

\[ Q_n \geq A_n \]

We prove this using the Cauchy-Schwarz inequality. For any set of positive real numbers \( a_1, a_2, \dots, a_n \), we have:

\[ \left( \sum_{i=1}^n a_i^2 \right) \cdot n \geq \left( \sum_{i=1}^n a_i \right)^2 \]

Dividing both sides by \( n^2 \) and taking the square root gives the QM-AM inequality:

\[ \sqrt{\frac{\sum_{i=1}^n a_i^2}{n}} \geq \frac{\sum_{i=1}^n a_i}{n} \]

General Proof Using Jensen's Inequality

Jensen's inequality provides a more general approach to proving inequalities of means. Let \( f(x) \) be a convex function. Then for any positive weights \( \lambda_1, \lambda_2, \dots, \lambda_n \) such that \( \lambda_1 + \lambda_2 + \dots + \lambda_n = 1 \), we have:

\[ f\left( \lambda_1 a_1 + \lambda_2 a_2 + \dots + \lambda_n a_n \right) \leq \lambda_1 f(a_1) + \lambda_2 f(a_2) + \dots + \lambda_n f(a_n) \]

For the specific case of \( f(x) = -\log(x) \), this leads to the AM-GM inequality. Similarly, using \( f(x) = x^2 \), we derive the QM-AM inequality.