Inequality

• Proofs of the AM-GM Inequality
  See several proofs of AM-GM, including an elementary two-number argument, a product-maximization proof, and a logarithm-based proof for n numbers.
• How to Prove the Quadratic Mean–Arithmetic Mean Inequality
  Prove that the quadratic mean is at least the arithmetic mean using Cauchy–Schwarz, and see when equality holds for nonnegative or real inputs.
• When Is $b^2c^2+c^2a^2+a^2b^2>abc(a+b+c)$ True?
  A sum-of-squares identity proves the expression is at least the right-hand side; equality occurs at a=b=c, so strictness needs an exception.
• Minimum of 1/a + 4/b + 9/c When a + b + c = 6
  Apply Cauchy–Schwarz to show the minimum is 6, attained at (a, b, c) = (1, 2, 3), which satisfies the positive-variable constraint.
• Is the AM-GM Inequality True for Four Numbers?
  For four nonnegative numbers, the arithmetic mean is at least the geometric mean. A short pairwise proof shows when equality holds.