Borell–Brascamp–Lieb inequality

The binomial approximation is useful for approximately calculating powers of numbers close to 1. It states that if $x$ is a real number close to 0 and $α$ is a real number, then

(1 + x)^{α} \approx 1 + α x .

This approximation can be obtained by using the binomial theorem and ignoring the terms beyond the first two.

The left-hand side of this relation is always greater than or equal to the right-hand side for $x > - 1$ and $α$ a non-negative integer, by Bernoulli's inequality.

Derivation using linear approximation

f (x) = (1 + x)^{α} .

f^{'} (x) = α (1 + x)^{α - 1} .

When x = 0:

f^{'} (0) = α .

Using linear approximation:

f (x) \approx f (a) + f^{'} (a) (x - a) .

f (x) \approx f (0) + f^{'} (0) (x - 0) .

(1 + x)^{α} \approx 1 + α x .

Derivation using Mellin transform

$M (p) = \int_{0}^{\infty} (1 + α x)^{- γ} x^{p - 1} d x$

Let

y = α x

$M (p) = α^{- p} \int_{0}^{\infty} (1 + y)^{- γ} y^{p - 1} d y$

Let

y = z / (1 - z)

$M (p) = α^{- p} \int_{0}^{1} (1 - z)^{γ - p - 1} z^{p - 1} d z$

$= α^{- p} B (γ - p, p)$

$= α^{- p} \frac{Γ (γ - p) Γ (p)}{Γ (γ)} .$

Using the inverse Mellin transform:

$(1 + α x)^{- γ} = \frac{1}{2 π i} \int_{c - i \infty}^{c + i \infty} (x α)^{- p} \frac{Γ (γ - p) Γ (p)}{Γ (γ)} d p$

Closing this integral to the left, which converges for $| α x | < 1$ , we get:

$(1 + α x)^{- γ} = Σ_{n = 0}^{\infty} (α x)^{n} \frac{(- 1)^{n}}{n!} \frac{Γ (γ + n)}{Γ (γ)}$

$= 1 - α x γ + (1 / 2) (α x)^{2} (γ + 1) γ - ...$

Borell–Brascamp–Lieb inequality

Derivation using linear approximation

Derivation using Mellin transform

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