AM-GM Inequality

The statement and the special case n = 2

For non-negative real numbers x₁, …, xₙ:

AM(x) = (x₁ + x₂ + … + xₙ) / n
GM(x) = (x₁ · x₂ · … · xₙ)^(1/n)

AM(x) ≥ GM(x),  with equality iff x₁ = x₂ = … = xₙ.

For n = 2, the statement becomes (a + b)/2 ≥ √(ab), or equivalently a + b ≥ 2√(ab). The proof in one line: from (√a − √b)² ≥ 0 (any squared real is non-negative) expand and rearrange. The slack — how much AM exceeds GM — is exactly

(a + b)/2 − √(ab) = (√a − √b)² / 2     for a, b ≥ 0

This is the cleanest closed form of the AM-GM gap. The square is zero only when √a = √b, i.e., a = b; otherwise it is strictly positive. Geometrically: arrange a and b along a number line; AM is the midpoint, GM is the side length of the square with the same area as the a × b rectangle. The Babylonian geometric mean is "what side length would a square need to match this rectangle's area?".

Proof via Jensen's inequality (concavity of log)

The function log : (0, ∞) → ℝ is strictly concave (its second derivative −1/x² is negative). By Jensen's inequality, for any non-negative weights wᵢ summing to 1 and any positive xᵢ:

log(Σ wᵢ xᵢ) ≥ Σ wᵢ log(xᵢ) = log(Π xᵢ^{wᵢ})

Exponentiate: Σ wᵢ xᵢ ≥ Π xᵢ^{wᵢ}. The equal-weight case wᵢ = 1/n gives AM ≥ GM. Equality in Jensen requires all xᵢ equal (since log is strictly concave). This proof handles the weighted version simultaneously, which is essential for downstream applications like Young's inequality.

To handle xᵢ = 0: if any xᵢ = 0 the GM is 0 and the AM is non-negative, so the inequality is trivial.

Proof via Cauchy's forward-backward induction

An elegant induction proof avoids Jensen. Step 1 (base): n = 2 from (√a − √b)² ≥ 0. Step 2 (doubling): if AM-GM holds for n = k, it holds for n = 2k by pairing:

AM_{2k}(x₁, …, x₂ₖ) = (AM_k(x₁, …, x_k) + AM_k(x_{k+1}, …, x_{2k})) / 2
                    ≥ √(AM_k(left) · AM_k(right))            [by n = 2 case]
                    ≥ √(GM_k(left) · GM_k(right))            [by induction]
                    = GM_{2k}(x₁, …, x_{2k}).

Step 3 (descent): given AM-GM holds for n, prove for n − 1. Take any x₁, …, x_{n−1} and set xₙ = AM(x₁, …, x_{n−1}). Then AM(x₁, …, xₙ) = AM(x₁, …, x_{n−1}), and AM-GM for n gives the same AM ≥ (Πxᵢ)^(1/n). Solve for the (n−1)-term GM and you recover AM-GM for n − 1. Cauchy's elegant 1821 trick: prove on powers of 2 first, then descend.

Worked examples with numbers

Example 1 (n = 2):
  a = 4, b = 9
  AM = (4 + 9)/2 = 6.5
  GM = √(4 · 9) = 6
  Slack = (√4 − √9)² / 2 = (2 − 3)² / 2 = 0.5
  6 ≤ 6.5    ✓  (strict; slack 0.5)

Example 2 (equality):
  a = b = 7
  AM = 7,  GM = 7
  7 = 7       EQUALITY (as expected)

Example 3 (n = 3):
  x = (1, 4, 8)
  AM = 13/3 ≈ 4.333
  GM = (32)^(1/3) ≈ 3.175
  4.333 ≥ 3.175    ✓

Example 4 (n = 4, equality):
  x = (5, 5, 5, 5)
  AM = 5,  GM = 5    EQUALITY

Example 5 (optimization — fixed sum 6, three terms):
  maximize xyz subject to x + y + z = 6, x, y, z ≥ 0
  AM-GM: (x + y + z)/3 = 2 ≥ (xyz)^(1/3)
  so xyz ≤ 8, attained iff x = y = z = 2.

Example 6 (weighted AM-GM):
  weights w = (1/3, 2/3), values x = (8, 1)
  weighted AM = (1/3)·8 + (2/3)·1 = 8/3 + 2/3 = 10/3 ≈ 3.333
  weighted GM = 8^(1/3) · 1^(2/3) = 2
  3.333 ≥ 2          ✓

The weighted AM-GM inequality

For non-negative weights w₁, …, wₙ summing to 1, and non-negative xᵢ:

w₁ x₁ + w₂ x₂ + … + wₙ xₙ  ≥  x₁^{w₁} · x₂^{w₂} · … · xₙ^{wₙ}

This is exactly Jensen's inequality applied to log on the convex combination. It generalizes the equal-weight case wᵢ = 1/n. The two-variable weighted form with w₁ = 1/p, w₂ = 1/q (where 1/p + 1/q = 1), x₁ = a^p, x₂ = b^q yields:

(1/p) a^p + (1/q) b^q ≥ (a^p)^{1/p} · (b^q)^{1/q} = ab

This is Young's inequality — the engine that drives Hölder's inequality. AM-GM is the grandfather of the L^p hierarchy.

AM-GM in the power-mean hierarchy

The power mean of order p is defined for non-negative xᵢ as

M_p(x) = ((1/n) Σ xᵢ^p)^(1/p)        for p ≠ 0
M_0(x) = lim_{p → 0} M_p(x) = (Π xᵢ)^(1/n)   (geometric mean)

The power-mean inequality says M_p is non-decreasing in p:

M_{−∞}(x) = min(xᵢ) ≤ HM = M_{−1} ≤ GM = M_0 ≤ AM = M_1 ≤ QM = M_2 ≤ … ≤ M_{∞} = max(xᵢ)

AM ≥ GM is the M_1 ≥ M_0 step in this chain. The general power-mean inequality is proved by combining AM-GM with Jensen applied to x ↦ x^(p/q). Equality holds throughout the chain iff all xᵢ are equal.

Common pitfalls

• Forgetting non-negativity. AM-GM requires all xᵢ ≥ 0. With negative inputs, the geometric mean may not be real, and the inequality breaks. For (−1, 1) the AM is 0 and the GM is undefined (or imaginary).
• Assuming equality without all-equal. AM = GM iff every xᵢ takes the same value. Two values being equal but a third different still gives strict AM > GM.
• Confusing weighted with unweighted. The standard AM-GM is the equal-weight case wᵢ = 1/n. Generalizing to unequal weights requires the weighted form Σwᵢ xᵢ ≥ Πxᵢ^{wᵢ}; the unweighted form is a special case.
• Using AM-GM where Cauchy-Schwarz is sharper. Sometimes the desired bound is squared norms; CS gives a tighter constant in those cases. AM-GM is the workhorse but not always the sharpest tool.
• Treating GM as easier to compute. The arithmetic mean is robust under noise; the geometric mean is sensitive to any zero (multiplied product becomes zero). Outlier behaviour differs.
• Forgetting "equality iff" means the equality case is a single point. The set where AM = GM is a measure-zero slice (the diagonal). For generic data, AM > GM strictly.

Where AM-GM shows up

• Optimization with sum-constraint. Maximizing the product of variables subject to a fixed sum is solved at the symmetric point by AM-GM. Used in inventory, pricing, and economic equilibrium models.
• Isoperimetric problems. Among all rectangles of fixed perimeter 2p, the square (sides p/2 each) has the maximum area p²/4 — direct AM-GM application. Generalizes to the continuous isoperimetric inequality.
• Young's inequality and L^p theory. Weighted AM-GM with weights 1/p, 1/q gives Young; Young drives Hölder; Hölder drives Minkowski. AM-GM is the foundational tier of L^p analysis.
• Concentration inequalities. Cauchy-Schwarz |⟨u, v⟩|² ≤ ‖u‖² · ‖v‖² is AM-GM applied to (‖u‖²‖v‖² ≥ |⟨u, v⟩|²). Jensen's inequality is the general framework.
• Information theory. Entropy bounds, Gibbs' inequality, and KL divergence non-negativity all rest on AM-GM (or equivalent log-concavity arguments).
• Finance — Kelly criterion. The Kelly betting strategy maximizes geometric mean of growth (long-run wealth) rather than arithmetic mean (single-period expected value). AM-GM quantifies the cost of volatility in compounding returns.
• Algorithmic mean computation. The arithmetic-geometric mean (AGM) algorithm iterates aₙ₊₁ = (aₙ + bₙ)/2, bₙ₊₁ = √(aₙ bₙ); convergence is quadratic, used in fast computation of π via Gauss-Legendre.
• Quantitative analysis. Volatility drag, expected vs. realized portfolio returns, geometric vs. arithmetic Sharpe ratios — every "averaging in time" problem in finance touches AM-GM.