- §1.1The Mean InequalitiesFour means in a row: harmonic ≤ geometric ≤ arithmetic ≤ quadratic.
- §1.2Proofs of the Mean InequalitiesKeep the sum, pull one number to the mean: the product never drops.
- §2.1Proving Inequalities with AM–GMWrite each small-side term as a weighted mean of big-side terms; add cyclically.
- §2.2Extrema with AM–GMPositive, fixed, equal: split terms to fix the product, then check equality.
- §2.3AM–GM in Geometric InequalitiesTangent lengths x, y, z sum to p; AM–GM on them, equality when equilateral.
- §2.4Variants of AM–GMReciprocals, squares, weights: a whole family, ordered by the power mean.
- §2.5AM–GM with ParametersPut in an unknown parameter, then choose it so all equality cases meet.
- §3.1The Cauchy Inequality and Its Proofs(Σaᵢbᵢ)² ≤ Σaᵢ²·Σbᵢ²; the gap is a squared area, zero only when flattened.
- §3.2Variants and Generalizations of CauchyMerging fractions into one can only lower the sum; equal ratios lose nothing.
- §4.1Proving Inequalities with CauchyBuild two lists whose pairwise products lose every variable, leaving a constant.
- §4.2Cauchy for Systems and ExtremaToo few equations? When Cauchy is forced to equality, the solution is pinned.
- §4.3Cauchy and Fractional InequalitiesMake each numerator a square, then merge once. What you multiply by decides it.
- §4.4Cauchy in Combinatorial EstimatesCount in two ways to bound Σdᵢ², then let Cauchy turn it into a bound on Σdᵢ.
- §4.5Cauchy with ParametersInsert a weight λ. Every λ gives a bound; one λ kills x and is reached.
- §4.6AM–GM and Cauchy TogetherChained bounds give the extremum only if every equality holds at the same point.