Inequality
The subject of mathematical inequalities is tied closely with optimization methods. While most of the subject of inequalities is often left out of the ordinary educational track, they are common in mathematics Olympiads.
Contents
Overview
Inequalities are arguably a branch of elementary algebra, and relate slightly to number theory. They deal with relations of variables denoted by four signs: .
For two numbers and :
- if is greater than , that is, is positive.
- if is smaller than , that is, is negative.
- if is greater than or equal to , that is, is either positive or .
- if is less than or equal to , that is, is either negative or .
Note that if and only if , , and vice versa. The same applies to the latter two signs: if and only if , , and vice versa.
Some properties of inequalities are:
- If , then , where .
- If , then , where .
- If , then , where .
Theorems
Here are some of the more useful inequality theorems, as well as general inequality topics.
- Arithmetic Mean-Geometric Mean Inequality
- Cauchy-Schwarz Inequality
- Carleman's Inequality
- Chebyshev's Inequality
- Geometric inequalities
- Hölder's inequality
- Isoperimetric inequalities
- Jensen's Inequality
- Maclaurin's Inequality
- Minkowski Inequality
- Muirhead's Inequality
- Nesbitt's Inequality
- Newton's Inequality
- Power mean inequality
- Ptolemy's Inequality
- Rearrangement Inequality
- Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
- Schur's Inequality
- Triangle Inequality
- Trivial inequality
Problems
Introductory
- Given that , show that . (<url>weblog_entry.php?t=172070 Source</url>)
Intermediate
- A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly . During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than . What's the largest number of matches she could've won before the weekend began? (Source)
Olympiad
- Let be positive real numbers. Prove that
(Source)
Resources
Books
Intermediate
Olympiad
- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.
- Inequalities by G. H. Hardy, J. E. Littlewood, G. Pólya.
Articles
Olympiad
- Inequalities by MIT Professor Kiran Kedlaya.
- Inequalities by IMO gold medalist Thomas Mildorf.
Classes
Olympiad
- The Worldwide Online Olympiad Training Program is designed to help students learn to tackle mathematical Olympiad problems in topics such as inequalities.