Proving the Inequality ( ab left( frac{1}{a} frac{1}{b} right) geq 4 ) using AM-GM and AM-HM Inequalities

In this article, we will dive deep into proving the inequality ( ab left( frac{1}{a} frac{1}{b} right) geq 4 ) using both the Arithmetic Mean-Geometric Mean (AM-GM) inequality and the Arithmetic Mean-Harmonic Mean (AM-HM) inequality. This is a fundamental proof that students often encounter and requires a good understanding of these inequalities.

Introduction to the Proof

The inequality in question is ( ab left( frac{1}{a} frac{1}{b} right) geq 4 ). This may initially appear complex, but it can be elegantly proven using the AM-GM inequality and the AM-HM inequality. We will explore both methods step by step.

Proof Using AM-GM Inequality

Let's start by rewriting the left-hand side of the inequality:

First, observe that

[ ab left( frac{1}{a} frac{1}{b} right) ab left( frac{b}{ab} frac{a}{ab} right) ab left( frac{b}{ab} frac{a}{ab} right) frac{ab^2 a^2b}{ab} b a a b. ]

Now, applying the AM-GM inequality to ( a ) and ( b ), we have:

[ frac{a b}{2} geq sqrt{ab}. ]

Squaring both sides, we get:

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

Which can be rewritten as:

[ frac{(a b)^2}{4} geq ab. ]

Therefore, multiplying both sides of this inequality by 4, we get:

[ (a b)^2 geq 4ab. ]

Since ( a b ab left( frac{1}{a} frac{1}{b} right) ), we can conclude:

[ ab left( frac{1}{a} frac{1}{b} right) geq 4. ]

Proof Using AM-HM Inequality

Alternatively, we can prove the inequality using the AM-HM inequality. The AM-HM inequality states that for any non-negative numbers ( x ) and ( y ),

[ frac{x y}{2} geq frac{2}{frac{1}{x} frac{1}{y}}. ]

Applying this to ( a ) and ( b ), we have:

[ frac{a b}{2} geq frac{2}{frac{1}{a} frac{1}{b}}. ]

Multiplying both sides by ( frac{1}{a} frac{1}{b} ), we get:

[ frac{a b}{2} left( frac{1}{a} frac{1}{b} right) geq 2. ]

Simplifying the left-hand side, we have:

[ frac{a b}{2a} frac{a b}{2b} frac{1}{2} left( frac{a}{a} frac{b}{a} frac{a}{b} frac{b}{b} right) frac{1}{2} left( 1 frac{a}{b} frac{b}{a} 1 right) frac{1}{2} left( 2 frac{a}{b} frac{b}{a} right) 1 frac{1}{2} left( frac{a}{b} frac{b}{a} right). ]

By the AM-GM inequality, we know:

[ frac{a}{b} frac{b}{a} geq 2. ]

Therefore:

[ 1 frac{1}{2} left( frac{a}{b} frac{b}{a} right) geq 1 frac{1}{2} cdot 2 2. ]

Multiplying both sides by 2, we get:

[ ab left( frac{1}{a} frac{1}{b} right) geq 4. ]

Conclusion

To summarize, we have shown that

[ ab left( frac{1}{a} frac{1}{b} right) geq 4 ]

using two different approaches: the AM-GM inequality and the AM-HM inequality. Both methods provide a clear and rigorous proof of the inequality. It is important to remember that the conditions ( a > 0 ) and ( b > 0 ) are necessary for the inequalities to hold.

Additional Notes on Proofs and Inequalities

In mathematics, it is crucial to understand the underlying principles and conditions when proving inequalities. The AM-GM and AM-HM inequalities are powerful tools in proving various mathematical statements. Familiarity with these inequalities can greatly enhance problem-solving skills and preparation for mathematical competitions.

Let's explore further with a detailed example:

Consider the expression ( ab left( frac{1}{a} frac{1}{b} right) ). We can break it down step-by-step to simplify the inequality:

[ ab left( frac{1}{a} frac{1}{b} right) ab left( frac{b}{ab} frac{a}{ab} right) ab left( frac{b a}{ab} right) frac{ab a^2 ab b^2}{ab} frac{2ab a^2 b^2}{ab}. ]

By the AM-GM inequality, we know:

[ frac{a b}{2} geq sqrt{ab}. ]

Therefore, squaring both sides, we get:

[ frac{a b}{2} geq sqrt{ab}. ]

Multiplying both sides by 4, we get:

[ 4 geq frac{(a b)^2}{ab}. ]

Thus:

[ 4ab geq (a b)^2. ]

Hence:

[ ab left( frac{1}{a} frac{1}{b} right) a b geq 4. ]