Proving the Inequality (a^2 b^2 geq 2ab) with Financial Mathematics

The inequality (a^2 b^2 geq 2ab) is a fundamental concept in mathematics, often used in various mathematical proofs and practical applications. This article will guide you through the process of proving this inequality and explore its relevance in financial mathematics, particularly in the context of interest and discount rates.

Proving the Inequality Algebraically

To begin, we start with the expression (a^2 b^2 geq 2ab). This expression can be rewritten as:

[a^2 b^2 - 2ab geq 0]

This can be further simplified by rearranging the terms:

[(a - b)^2 geq 0]

Since the square of any real number is non-negative, ((a - b)^2 geq 0) is always true. Therefore, we have:

[a^2 b^2 geq 2ab]

Alternative Proof: Simplifying the Expression

Another way to prove this inequality is by manipulating the terms and using algebraic identities. Let's consider the expression:

[a^2 b^2]

We can rewrite this expression as:

[a^2 b^2 a^2 b^2 - 2ab 2ab]

This can be factored as:

[a^2 b^2 (a - b)^2 2ab]

Since ((a - b)^2 geq 0) and (2ab geq 0) for all real numbers (a) and (b), it follows that:

[a^2 b^2 geq 2ab]

Financial Mathematics Perspective

Now, let's explore the context of this inequality in financial mathematics. Consider the situation where (a) represents the present value and (b) represents the future value of an investment. Let's break this down into two parts:

Part 1: Algebraic Proof in Financial Context

In a financial context, we can rewrite the inequality as:

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

Adding (2ab) to the original inequality, we get:

[a^2 b^2 ab ab - 2ab]

Dividing by (ab), we get:

[frac{a^2}{ab} frac{b^2}{ab} - 2 frac{a}{b} frac{b}{a} - 2]

This simplifies to:

[left|frac{a}{b} frac{b}{a} - 2right| 2]

Therefore:

[left|frac{a b - 2ab}{ab}right| 2]

Part 2: Interest and Discount Rates

Let's introduce the concepts of interest and discount rates. Suppose (b) is 5 more than (a), where (b) represents the future value after one year of investment at a 5% interest rate. Using the formula for future value, we have:

[frac{b}{a} frac{1.05a}{a} 1.05]

The reciprocal of this, representing the present value factor, is:

[frac{a}{b} frac{a}{1.05a} frac{1}{1.05} approx 0.9524]

The difference from 1 is the discount rate (d), which is:

[d 1 - frac{1}{1.05} 1 - 0.9524 0.0476 approx 4.76%]

Now, considering the inequality in financial terms, we can write:

[left| frac{b}{a} frac{a}{b} - 2 right| left| 1.05 0.9524 - 2 right| left| 2.002 right| 2]

This demonstrates that the inequality holds in financial contexts, reflecting the relationship between interest rates and discount rates.

Simpler Cases

Let's consider a simpler case where (a 100) and (b 80). In this scenario, the future value is 100, and the present value is 80, with an interest rate (i 0.25). The discount rate (d) is:

[d 1 - frac{1}{1 i} 1 - frac{1}{1.25} 1 - 0.8 0.2]

So, we have:

[left| frac{b}{a} frac{a}{b} - 2 right| left| frac{80}{100} frac{100}{80} - 2 right| left| 0.8 1.25 - 2 right| left| 2.05 - 2 right| 0.05]

This illustrates that the inequality still holds, even in simpler cases, with the difference reflecting the small shortfalls or gains in financial terms.

Conclusion

The inequality (a^2 b^2 geq 2ab) is not only a mathematical truth but also has practical applications in financial mathematics. By understanding this inequality, we gain insights into the relationship between present and future values, interest rates, and discount rates. This knowledge is invaluable in various financial scenarios, helping us make better investment decisions.