A
power inequality is an inequality containing terms of the form
ab, where
a and
b are real positive numbers or variable expressions. They often appear in
mathematical olympiads exercises. Examples: • For any real
x, e^x \ge 1+x. • If
x > 0 and
p > 0, then \frac1p\left(x^p - 1\right) \ge \ln(x) \ge \frac1p\left(1 - \frac{1}{x^p}\right). In the limit of
p → 0, the upper and lower bounds converge to ln(
x). • If
x > 0, then x^x \ge \left( \frac{1}{e}\right)^\frac{1}{e}. • If
x > 0, then x^{x^x} \ge x. • If
x,
y,
z > 0, then \left(x+y\right)^z + \left(x+z\right)^y + \left(y+z\right)^x > 2. • For any real distinct numbers
a and
b, \frac{e^b-e^a}{b-a} > e^\frac{a+b}{2}. • If
x,
y > 0 and 0 x^p+y^p > \left(x+y\right)^p. • If
x,
y,
z > 0, then x^x y^y z^z \ge \left(xyz\right)^\frac{x+y+z}{3}. • If
a,
b > 0, then a^a + b^b \ge a^b + b^a. • If
a,
b > 0, then a^{ea} + b^{eb} \ge a^{eb} + b^{ea}. • If
a,
b,
c > 0, then a^{2a} + b^{2b} + c^{2c} \ge a^{2b} + b^{2c} + c^{2a}. • If
a,
b > 0, then a^b + b^a > 1. == Well-known inequalities ==