The Bernoulli inequality in mathematics states that for real numbers x>-1,

When n≥1, (1+x)ⁿ≥1+nx holds;

When 0≤n≤1, (1+x)ⁿ≤1+nx holds true.

You can see that the equality holds if and only if n = 0, 1, or x = 0. Bernoulli's inequality is often used as a key step in proving other inequalities.

The general form of Bernoulli's inequality is (1+x1+x2+x3···+xn)<=(1+x1)(1+x2)(1+x3)···(1+xn), (for any 1 <= i, j <= n, there is xi >= -1 and sign(xi) = sign(xj), that is, all xi have the same sign and are greater than or equal to -1) The equality holds if and only if n=1

Note: The letters or numbers after x are subscripts