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 $\alpha$ is a real number, then

$(1+x{)}^{\alpha }\approx 1+\alpha x.$

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 $\alpha$ a non-negative integer, by Bernoulli's inequality.

Derivation using linear approximation

$f(x)=(1+x{)}^{\alpha }.$

$f^{\prime }(x)=\alpha (1+x{)}^{\alpha -1}.$

When x = 0:

$f^{\prime }(0)=\alpha .$

Using linear approximation:

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

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

$(1+x{)}^{\alpha }\approx 1+\alpha x.$

Derivation using Mellin transform

$M(p)=\underset{0}{\overset{\mathrm{\infty }}{\int }}(1+\alpha x{)}^{-\gamma }{x}^{p-1}dx$

Let $y=\alpha x$

$M(p)={\alpha }^{-p}\underset{0}{\overset{\mathrm{\infty }}{\int }}(1+y{)}^{-\gamma }{y}^{p-1}dy$

Let $y=z/(1-z)$

$M(p)={\alpha }^{-p}\underset{0}{\overset{1}{\int }}(1-z{)}^{\gamma -p-1}{z}^{p-1}dz$

$={\alpha }^{-p}B(\gamma -p,p)$

$={\alpha }^{-p}\frac{\mathrm{\Gamma }(\gamma -p)\mathrm{\Gamma }(p)}{\mathrm{\Gamma }(\gamma )}.$

Using the inverse Mellin transform:

$(1+\alpha x{)}^{-\gamma }=\frac{1}{2\pi i}\underset{c-i\mathrm{\infty }}{\overset{c+i\mathrm{\infty }}{\int }}(x\alpha {)}^{-p}\frac{\mathrm{\Gamma }(\gamma -p)\mathrm{\Gamma }(p)}{\mathrm{\Gamma }(\gamma )}dp$

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

$(1+\alpha x{)}^{-\gamma }={\mathrm{\Sigma }}_{n=0}^{\mathrm{\infty }}(\alpha x{)}^n\frac{(-1{)}^n}{n!}\frac{\mathrm{\Gamma }(\gamma +n)}{\mathrm{\Gamma }(\gamma )}$

$=1-\alpha x\gamma +(1/2)(\alpha x{)}^2(\gamma +1)\gamma -...$