How Do I Find All Positive Integer Pairs Using H?lder's Inequality?

In this article, we will explore how to use H?lder's Inequality to find all positive integer pairs that satisfy a given condition. This method is particularly useful in mathematical optimization and number theory problems. We will begin by understanding H?lder's Inequality and then apply it to a specific problem involving positive integers.

Understanding H?lder's Inequality

H?lder's Inequality is a fundamental inequality in mathematical analysis, named after the Finnish mathematician Eliakim Hastings Moore, who introduced it under the German name H?lder (handbuch, §80).

The general form of H?lder's Inequality states that if we have non-negative sequences of real numbers {aj}j1n, {bj}j1n, ..., {zj}j1n, and corresponding weights {λa, λb, ..., λz} such that each λx ≥ 0 and their sum is 1, then:

Big(sum_{i1}^n a_iBig)^{lambda_a} Big(sum_{i1}^n b_iBig)^{lambda_b} ... Big(sum_{i1}^n z_iBig)^{lambda_z} geq sum_{i1}^n a_i^{lambda_a} b_i^{lambda_b} ... z_i^{lambda_z}

Equality holds when all the sequences are directly proportional.

Applying H?lder's Inequality to Find Positive Integer Pairs

Let's consider the problem of finding all positive integer pairs (a, b) that satisfy the following inequality:

1 a 8b ≥ ba b

We will use H?lder's Inequality to solve this problem.

Step 1: Define the Sequences and Weights

First, we define the sequences and their corresponding weights:

Sequences: {1, a}, {b, 8}, {a, b} Weights: λ1 λ2 λ3 1/3

Step 2: Apply H?lder's Inequality

Applying H?lder's Inequality, we get:

(1 a 8b)^(1/3) (b 8)^(1/3) (a b)^(1/3) ≥ (1^(1/3) * (b * a^(1/3)) * (8 * b^(1/3))

After simplifying, this becomes:

(1 a 8b)^(1/3) (b 8)^(1/3) (a b)^(1/3) ≥ b * a^(1/3) * 2

Now, we cube both sides to remove the fractional exponents:

1 a 8b ≥ 27ab

Step 3: Determine the Equality Case

The equality case of H?lder's Inequality occurs when the sequences are directly proportional. Therefore, we need to ensure that:

1/b b/8 a/8b

From these proportions, we get:

b^2 8

b 2√2 (not applicable since b must be a positive integer)

However, if we assume b 2, then a 4 satisfies the equality case:

a 8b 1 a 8b 27ab

Thus:

(2 8 * 2) 1 2 8 * 2 27 * 2 * 4 216

Square both sides:

(1 2 8 * 2) 216

Therefore, the only positive integer pair (a, b) that satisfies this condition is:

(a, b) (2, 4)

Conclusion

In this article, we have demonstrated how to use H?lder's Inequality to find the positive integer pairs (a, b) that satisfy the given inequality. The method involves defining sequences, applying the inequality, and ensuring the equality case holds. The result is the unique positive integer pair (a, b) (2, 4).

Understanding and applying such inequalities in mathematical problems is crucial in various fields, including optimization, number theory, and more. By mastering these techniques, one can solve complex problems efficiently and accurately.