Toolbox

A collection of commonly used methods, inequalities, and tools in mathematical research.

General Methods

• Pigeonhole Principle

  If $n$ pigeons fly into $m$ pigeonholes and $n > m$, then at least one pigeonhole contains no fewer than two pigeons. Commonly used in existence proofs and combinatorial counting.

• Proof by Contradiction

  Assume the proposition to be proved is false, and derive a contradiction with known conditions or axioms, thereby proving the original proposition. Applicable to negative propositions and uniqueness proofs.

• Mathematical Induction

  Used to prove propositions related to natural numbers: verify the base case holds, and assume the statement holds for $n$ to derive that it also holds for $n+1$.

• Constructive Method

  Proves existence by directly constructing an object that satisfies the given conditions. Commonly used in recurrence relations, interpolation formulas, and the construction of special functions.

Common Inequalities

• Hölder's Inequality

  Let $p, q > 1$ with $\frac{1}{p} + \frac{1}{q} = 1$. Then for measurable functions $f, g$:

  $$\int |fg| \, d\mu \leq \left(\int |f|^p \, d\mu\right)^{\frac{1}{p}} \cdot \left(\int |g|^q \, d\mu\right)^{\frac{1}{q}}$$

  A core tool in $L^p$ space theory, widely used in partial differential equations and functional analysis.

• Cauchy-Schwarz Inequality

  In an inner product space:

  $$|\langle u, v \rangle| \leq \|u\| \cdot \|v\|$$

  A special case of Hölder's inequality when $p = q = 2$, one of the most commonly used inequalities in estimation and bounding.

• Minkowski Inequality

  The triangle inequality for the $L^p$ norm ($p \geq 1$):

  $$\|f + g\|_p \leq \|f\|_p + \|g\|_p$$

  The foundation for proving that $L^p$ spaces are normed linear spaces.

• Young's Inequality

  For $a, b \geq 0$ and conjugate exponents $p, q$:

  $$ab \leq \frac{a^p}{p} + \frac{b^q}{q}$$

  A key lemma for proving Hölder's inequality, frequently used in integral estimation and convex analysis.

• Gronwall's Inequality

  If $u(t) \leq a + \int_0^t b(s)u(s)\, ds$, then:

  $$u(t) \leq a \cdot \exp\!\left(\int_0^t b(s)\, ds\right)$$

  A core tool for uniqueness and stability analysis of solutions to differential and integral equations.

• Jensen's Inequality

  For a convex function $\varphi$:

  $$\varphi(\mathbb{E}[X]) \leq \mathbb{E}[\varphi(X)]$$

  A fundamental inequality in probability theory and convex analysis, commonly used in expectation estimation and information theory.