The Arithmetic Mean-Geometric Mean Inequality: Proving the Bound for Positive Real Numbers

The Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality) is a fundamental theorem in mathematical analysis. It states that for a set of positive real numbers a_i, the arithmetic mean of these numbers will always be greater than or equal to their geometric mean:

RHS left( frac{sum_{i1}^{n} a_i}{n} right) geq left( prod_{i1}^{n} a_i right)^{frac{1}{n}} LHS

Understanding the AM-GM Inequality

Consider a set of n positive real numbers denoted as a_1, a_2, ..., a_n. The arithmetic mean (AM) of these numbers is given by:

AM frac{a_1 a_2 ... a_n}{n}

The geometric mean (GM) is given by:

GM left( a_1 cdot a_2 cdot ... cdot a_n right)^{frac{1}{n}}

The AM-GM Inequality asserts that for positive real numbers, the arithmetic mean is always greater than or equal to the geometric mean. This relationship is often denoted as:

frac{a_1 a_2 ... a_n}{n} geq left( a_1 cdot a_2 cdot ... cdot a_n right)^{frac{1}{n}}

Proof of the AM-GM Inequality for Positive Real Numbers

To prove the AM-GM Inequality, we can start from two basic cases:

Case 1: Two Positive Real Numbers

For two positive real numbers a and b, the AM-GM Inequality is proven as follows:

frac{a b}{2} geq sqrt{ab}

Subtracting sqrt{ab} from both sides:

frac{a b - 2sqrt{ab}}{2} geq 0

Factoring the left-hand side:

frac{sqrt{a}^2 - 2sqrt{a}sqrt{b} sqrt{b}^2}{2} frac{(sqrt{a} - sqrt{b})^2}{2} geq 0

Since the square of any real number is non-negative, the inequality holds true.

Case 2: General Case with n Positive Real Numbers

The AM-GM Inequality for n positive real numbers a_1, a_2, ..., a_n can be proven using mathematical induction.

The base case (n 2) has already been proven. Now, assume the inequality holds for n k positive real numbers:

frac{a_1 a_2 ... a_k}{k} geq left( a_1 cdot a_2 cdot ... cdot a_k right)^{frac{1}{k}}

We need to prove that it holds for n k 1. Consider the following:

frac{a_1 a_2 ... a_k a_{k 1}}{k 1} frac{ka_1 a_{k 1}}{k 1} cdot frac{1}{k} frac{a_2 a_3 ... a_k}{k} cdot frac{1}{k}

By the inductive hypothesis, we have:

frac{a_2 a_3 ... a_k}{k} geq left( a_2 cdot a_3 cdot ... cdot a_k right)^{frac{1}{k-1}}

Multiplying both sides by frac{1}{k}:

frac{a_2 a_3 ... a_k}{k} geq left( a_2 cdot a_3 cdot ... cdot a_k right)^{frac{1}{k}} cdot frac{1}{k}

Adding this to the other part:

frac{a_1 a_2 ... a_k a_{k 1}}{k 1} geq left( a_1 cdot left( a_2 cdot a_3 cdot ... cdot a_k right)^{frac{1}{k-1}} cdot a_{k 1} right)^{frac{1}{k 1}}

This completes the inductive step and proves the AM-GM Inequality for any positive integer n.

Conclusion

The AM-GM Inequality is a powerful tool in mathematical analysis and optimization. It guarantees that for any set of positive real numbers, the arithmetic mean will always be greater than or equal to the geometric mean. This inequality holds true because it is based on the fundamental properties of squares and square roots of real numbers.

It is important to note that the equality holds if and only if all the numbers in the set are equal. When any of the numbers are negative or zero, the inequality may not hold, as demonstrated by the counterexample:

a b c d -1

a cdot b cdot c cdot d cdot frac{1}{abcd} -3 cdot 5

However, the AM-GM Inequality strictly applies to positive real numbers, ensuring a robust mathematical framework for various applications.