About

Questions on proving, manipulating and applying inequalities.

An inequality is a mathematical relation between two quantities that are not necessarily equal, but bigger or smaller.

To prove inequalities, a number of proven inequalities can be used, including:

  • The AM-GM inequality, that states that for any non-negative real numbers $a_1, a_2, a_3, \cdots, a_n$ we have:

$$\frac{a_1+a_2+a_3+\cdots+a_n}{n} \ge \sqrt[n]{a_1a_2a_3\cdots a_n}$$

  • The generalized mean inequality, that states that for any positive real numbers $a_1, a_2, a_3, \cdots, a_n$ and real $p>q$ the following inequality is true:

$$\left(\frac{a_1^p+a_2^p+a_3^p+\cdots+a_n^p}{n}\right)^{\frac{1}{p}} > \left(\frac{a_1^q+a_2^q+a_3^q+\cdots+a_n^q}{n}\right)^{\frac{1}{q}}$$

  • The rearrangement inequality
  • The Cauchy-Schwarz inequality: For any real numbers $a_1, a_2, a_3, \cdots, a_n$, $b_1, b_2, b_3, \cdots, b_n$ the following inequality is true: $$\left(\sum^{n}_{i=1} a_ib_i \right)^2 < \left(\sum^{n}_{i=1} a_i^2 \right)\left(\sum^{n}_{i=1} b_i^2\right)$$
  • The Schur inequalities: For nonnegative real numbers $x,y,z$ and positive number $t$, the following inequality is true:$$x^t(x-y)(x-z)+y^t(y-z)(y-x)+z^t (z-x)(z-y)\ge 0$$
  • Muirhead inequalities
history | excerpt history