Sharp and Non-Sharp Inequalities: An Advanced Example

1. Sharp Inequality Example: The Arithmetic Mean-Geometric Mean Inequality (AM-GM)

The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any non-negative real numbers $x_1, x_2, \dots, x_n$, the following inequality holds:

$\frac{x_1 + x_2 + \dots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \dots x_n}$ with equality if and only if $x_1 = x_2 = \dots = x_n$.

Sharpness of the AM-GM Inequality

• The AM-GM inequality is sharp because equality holds when all the variables are equal, i.e., when $x_1 = x_2 = \dots = x_n$.
• This means the inequality cannot be improved in any way—there is no smaller upper bound for the geometric mean given the arithmetic mean.
• Example: For $x_1 = 4$ and $x_2 = 4$, the AM-GM inequality gives: $\frac{4 + 4}{2} = 4 \geq \sqrt{4 \times 4} = 4$ Equality holds here, making the AM-GM inequality sharp for this case.

2. Non-Sharp Inequality Example: The Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality for vectors $\mathbf{u} = (u_1, u_2, \dots, u_n)$ and $\mathbf{v} = (v_1, v_2, \dots, v_n)$ states that:

$\left( \sum_{i=1}^{n} u_i v_i \right)^2 \leq \left( \sum_{i=1}^{n} u_i^2 \right) \left( \sum_{i=1}^{n} v_i^2 \right)$

Equality holds if and only if $\mathbf{u}$ and $\mathbf{v}$ are linearly dependent, meaning that $\mathbf{u} = \lambda \mathbf{v}$ for some scalar $\lambda$.

Non-Sharpness of the Cauchy-Schwarz Inequality

• The Cauchy-Schwarz inequality is not sharp in the sense that it does not give a tight bound for the left-hand side in cases where the vectors $\mathbf{u}$ and $\mathbf{v}$ are not proportional.

• In other words, there is potential for improvement or refinement of the inequality in certain cases, especially when the vectors are not linearly dependent.

• Example: For $\mathbf{u} = (1, 2)$ and $\mathbf{v} = (2, 1)$, we compute:

  $\left( 1 \times 2 + 2 \times 1 \right)^2 = (2 + 2)^2 = 16$ $\left( 1^2 + 2^2 \right) \left( 2^2 + 1^2 \right) = (1 + 4)(4 + 1) = 5 \times 5 = 25$

  The inequality holds: $16 \leq 25$

  But here, the inequality is not tight because $\mathbf{u}$ and $\mathbf{v}$ are not proportional, so the bound is not sharp in this case.

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Key Takeaways:

• Sharp inequalities like the AM-GM inequality give tight bounds where equality holds under specific conditions.
• Non-sharp inequalities like Cauchy-Schwarz may provide valid bounds, but the bounds are not always as tight, and equality holds under more restrictive conditions (e.g., linear dependence).

Would you like further details or more examples on these concepts?

Here are 5 follow-up questions:

1. How can the AM-GM inequality be generalized for more than two variables?
2. What are some applications of the Cauchy-Schwarz inequality in optimization problems?
3. How does the concept of sharpness apply to other inequalities like Hölder's or Minkowski's inequality?
4. Can sharpness be improved or refined in non-sharp inequalities under certain conditions?
5. What is the role of symmetry in determining whether an inequality is sharp?

Tip: Always look for specific conditions that trigger equality in inequalities. Sharp inequalities often have clear criteria for equality, which can simplify problem-solving!