Proving Inequalities Involving Real Numbers

In this article, we explore the problem of proving inequalities involving real numbers. Specifically, we will show that for real numbers x and y satisfying certain conditions, the inequality xy(x^4y^3 / (3xy - y - 1) ≥ 1) holds. This problem is a great example of how algebraic manipulation and inequalities can be used to derive rigorous proofs.

Problem Statement and Constraints

Given that (x) and (y) are real numbers satisfying the following conditions:

The goal is to show that the inequality (frac{x^4y^3}{3xy - y - 1} geq 1) holds.

Initial Observations

First, we observe that if (x y 0), the inequality becomes (0 geq 1), which is clearly not true. Therefore, we must have (xy in mathbb{R} setminus {0}).

Exploring the Constraints

From the first constraint, (xy > 0), we know that at most one of (x) and (y) can be negative. Without loss of generality, let's assume (x geq 0). Using the second constraint, we can derive more information about (y).

Deriving Boundaries for (y)

From the second constraint, we have (4x(1 - y) leq -y^2). Since (x geq 0), we can divide by (4x) (which is positive). This gives us (1 - y leq -frac{y^2}{4x}), and by rearranging, we get (y(1 frac{y^2}{4x}) geq 1). Further simplifying, we find that (y geq 1).

Implications on the Third Constraint

Using the third constraint (x^2 1 leq frac{y^2}{2}), we substitute (y geq 1) and find that ((1 - x^2) / 2 geq 1). This is only possible if (x^2 leq 1/2). Therefore, (x geq 1).

Substituting (x geq 1) into the third constraint, we get (y geq 2).

Proving the Inequality

Now that we have (x geq 1) and (y geq 2), we need to establish that (frac{x^4y^3}{3xy - y - 1} geq 1). We can start by showing that the numerator is greater than or equal to the denominator:

[x^4y^3 - 3xy - y - 1 geq 0]

By factoring, we get:

[x^4y^3 - 3xy - y - 1 (x^4 - 1)y^3 - y - 3xy geq 0]

Since (x^4 - 1 geq 0) and (4xy leq 4x^2y^2), we have:

[(x^4 - 1)y^3 - y - frac{3}{4} cdot 4x^2y^2 geq 0]

Simplifying further:

[y^3 - y - frac{3}{4}(4x^2y^2) geq 0]

[y^3 - frac{3}{4}y^2 - y - frac{3}{2}y^2 geq 0]

[y^3 - frac{9}{4}y^2 - y geq 0]

This can be written as:

[frac{1}{4}(4y^3 - 9y^2 - 4y 12) geq 0]

The expression (4y^3 - 9y^2 - 4y 12) can be factored as ((4y - 3)(y - 2)^2), which is non-negative for (y geq 2), given our constraints.

Conclusion

Thus, we have shown that for real numbers (x) and (y) satisfying the given conditions, the inequality (frac{x^4y^3}{3xy - y - 1} geq 1) holds. This example highlights the power of algebraic manipulation and the importance of carefully analyzing constraints to derive rigorous proofs.