Inequalities

Classical and analytic inequalities at the heart of competition math and analysis.

Triangle Inequality (Norms) — x + y ≤ x + y — the norm axiom behind absolute value, Euclidean length, and distance.
AM-GM — a+b/2 ≥ √ab — bounds for products via sums and back.
Cauchy-Schwarz — ( a · b)^2 ≤ | a|^2 | b|^2 — the universal vector inequality.
Jensen's Inequality — Convex f: f(E[X]) ≤ E[f(X)].
Power Mean — M_r ≤ M_s for r ≤ s — one inequality unifies HM, GM, AM, and QM.
Rearrangement — Big times big beats big times small — sorted pairs maximize ∑ a_i b_ (i).
Schur's Inequality — a^3 + b^3 + c^3 + 3abc ≥ ab(a+b) + bc(b+c) + ca(c+a) — tight at a=b=c and at (a,a,0).
Muirhead's Inequality — More spread-out exponents give bigger symmetric sums: if a b then [ a] ≥ [ b].
Sum of Squares (SOS) — f = ∑ g_i^2 h_i ≥ 0 — the universal hammer for symmetric inequalities.
Hölder's Inequality — ∑ |a_i b_i| ≤ |a|_p |b|_q for conjugate p, q — Cauchy–Schwarz generalized to L^p.
Minkowski's Inequality — |a + b|_p ≤ |a|_p + |b|_p — the triangle inequality in every L^p.

Further Topics

Chebyshev's Sum Inequality — For similarly sorted sequences, the mean of products beats the product of means.
Karamata's Inequality (Majorization) — Convexity plus majorization: the inequality that generalizes Jensen.
Young's Inequality — Bound a product by a sum of powers using conjugate exponents.
Bernoulli's Inequality — The bound (1+x)^n ≥ 1 + nx, its induction proof, and how it seeds AM-GM.
Tangent Line Trick — Bound a convex summand by its tangent line, then sum.
Abel Summation — Summation by parts: trade a sum of products for differences of one factor.
Schur-Convexity — Majorization, Schur-convex functions, and the Schur-Ostrowski criterion.
Hilbert's Inequality — The double-sum bound with the sharp constant pi.
Ravi Substitution — Trade triangle sides for positive reals and the constraint vanishes.
Smoothing / Mixing Variables — Nudge variables toward equality without worsening the objective; the extremum sits at the balanced point.