Common Mathematical Inequalities

Why This Matters

Mathematical inequalities aren't just abstract formulas to memorize—they're the workhorses that let mathematicians establish bounds, prove theorems, and solve optimization problems. When you're working through calculus, linear algebra, or probability, you'll constantly need to show that one quantity is larger or smaller than another. These inequalities give you the tools to do exactly that. You're being tested on your ability to recognize when to apply each inequality, understand the conditions required for each to hold, and connect them to broader mathematical structures like norms, convexity, and expected values.

Think of inequalities as a toolkit: some handle geometric relationships, others tame products and sums, and still others give you control over random variables. The key is understanding which tool fits which job. Don't just memorize the formulas—know what mathematical principle each inequality captures and when its conditions are satisfied. That conceptual understanding will serve you far better on exams than rote recall.

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Geometric and Metric Inequalities

These inequalities establish fundamental properties of distance and magnitude. They're the foundation for understanding how distances behave in mathematical spaces.

Triangle Inequality

• Absolute values satisfy $|a + b| \leq |a| + |b|$—the "length" of a sum never exceeds the sum of lengths
• Geometric interpretation: in any triangle, the sum of two side lengths must exceed the third side
• Foundation for metrics—any valid distance function must satisfy this property, making it essential for analysis

Minkowski's Inequality

• Generalizes the triangle inequality to $L^p$ spaces: $||x + y||_p \leq ||x||_p + ||y||_p$
• Works for any $p \geq 1$, where the $p$-norm is defined as $||x||_p = \left(\sum |x_i|^p\right)^{1/p}$
• Proves that $L^p$ spaces are valid vector spaces—without this inequality, we couldn't define proper distances in these function spaces

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Product and Sum Bounding Inequalities

These inequalities help you control products by relating them to sums—crucial when you need to bound a complicated product using simpler terms.

Cauchy-Schwarz Inequality

• The workhorse of linear algebra: $(a_1b_1 + \cdots + a_nb_n)^2 \leq (a_1^2 + \cdots + a_n^2)(b_1^2 + \cdots + b_n^2)$
• Equivalently written using inner products as $|\langle u, v \rangle|^2 \leq \langle u, u \rangle \cdot \langle v, v \rangle$
• Equality holds exactly when vectors are parallel—use this condition to identify when bounds are tight

Hölder's Inequality

• Generalizes Cauchy-Schwarz to conjugate exponents: $\sum |a_i b_i| \leq \left(\sum |a_i|^p\right)^{1/p} \left(\sum |b_i|^q\right)^{1/q}$ where $\frac{1}{p} + \frac{1}{q} = 1$
• Cauchy-Schwarz is the special case $p = q = 2$—recognizing this connection helps you choose the right tool
• Essential for functional analysis and proving integrability results in $L^p$ spaces

Young's Inequality

• Bounds products by sums of powers: $ab \leq \frac{a^p}{p} + \frac{b^q}{q}$ for non-negative $a, b$ and conjugates $\frac{1}{p} + \frac{1}{q} = 1$
• Often used to prove Hölder's inequality—it's lower-level in the hierarchy of tools
• Key technique: when you see a product you need to bound, try splitting it using Young's with strategic exponent choices

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Mean and Averaging Inequalities

These inequalities relate different types of averages and reveal how convexity shapes the behavior of functions applied to means.

Arithmetic Mean-Geometric Mean (AM-GM) Inequality

• For non-negative reals: $\frac{x_1 + x_2 + \cdots + x_n}{n} \geq (x_1 x_2 \cdots x_n)^{1/n}$—arithmetic mean beats geometric mean
• Equality holds when all values are equal—this is your key to optimization problems
• Go-to tool for optimization: when maximizing a product subject to a sum constraint (or vice versa), AM-GM often gives the answer directly

Jensen's Inequality

• For convex functions $f$: $f\left(\frac{x_1 + \cdots + x_n}{n}\right) \leq \frac{f(x_1) + \cdots + f(x_n)}{n}$
• Inequality reverses for concave functions—always check the curvature before applying
• Explains why AM-GM works: since $-\ln(x)$ is convex, Jensen's applied to logarithms yields AM-GM

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Growth and Approximation Inequalities

This inequality captures how exponential-type growth behaves, particularly useful for approximations and induction proofs.

Bernoulli's Inequality

• For $x \geq -1$ and integer $n \geq 0$: $(1 + x)^n \geq 1 + nx$
• Provides a linear lower bound on exponential growth—the bound is tight when $x$ is small
• Perfect for induction proofs and establishing limits; often appears when bounding $(1 + \frac{1}{n})^n$ type expressions

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Probabilistic Bounding Inequalities

These inequalities let you control probabilities without knowing the full distribution—essential for worst-case analysis in statistics and probability.

Markov's Inequality

• For non-negative random variable $X$ and $a > 0$: $P(X \geq a) \leq \frac{E[X]}{a}$
• Requires only the expected value—no variance or distribution shape needed
• Often gives weak bounds but works universally; it's your baseline tool when you know almost nothing about $X$

Chebyshev's Inequality

• Bounds deviation from the mean: $P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}$
• Uses variance information to give tighter bounds than Markov—the tradeoff is you need to know $\sigma^2$
• Distribution-free guarantee: no matter the shape, at least $1 - \frac{1}{k^2}$ of probability lies within $k$ standard deviations

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Quick Reference Table

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Self-Check Questions

1. Both Cauchy-Schwarz and Hölder's inequality bound sums of products. What condition on the exponents makes Cauchy-Schwarz a special case of Hölder's?

2. You're given a non-negative random variable with known mean but unknown variance. Which inequality can you use to bound $P(X \geq a)$, and what would you need to know to get a tighter bound?

3. Compare and contrast AM-GM and Jensen's inequality: How is AM-GM derived from Jensen's, and when would you prefer one over the other?

4. The triangle inequality and Minkowski's inequality both establish that "the norm of a sum is at most the sum of norms." What distinguishes their domains of application?

5. If an exam problem asks you to prove that a product $ab$ is bounded above by a sum of powers of $a$ and $b$, which inequality should you reach for first, and what condition must the exponents satisfy?