Difference between revisions of "Inequality"
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*If <math>a \ge b</math>, then <math>a+c>b</math>, where <math>c>0</math>. | *If <math>a \ge b</math>, then <math>a+c>b</math>, where <math>c>0</math>. | ||
| − | == | + | ==Common theorems== |
| − | * [[ | + | * The [[Arithmetic Mean-Geometric Mean Inequality]], its extension, the [[Power Mean Inequality]], and its other extension, the [[Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality]]. |
| − | + | * The [[Trivial Inequality]]. | |
| − | * [[Trivial Inequality]] | + | * For others, consult [[User:Temperal/The Problem Solver's Resource4|this page]], [[User:Temperal/The Problem Solver's Resource8|this page]], and [[User:Temperal/The Problem Solver's Resource11|this page]]. |
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==Intermediate== | ==Intermediate== | ||
Revision as of 18:32, 25 October 2007
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 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 .
is positive.
if 
.
if 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
.
, where 
.
, then 
, where 
.Common theorems
- The Arithmetic Mean-Geometric Mean Inequality, its extension, the Power Mean Inequality, and its other extension, the Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality.
- The Trivial Inequality.
- For others, consult this page, this page, and this page.
Intermediate
Example Problems
Olympiad
See the list of famous inequalities below
Famous inequalities
Here are some of the more famous and useful inequalities, as well as general inequalities topics.
- Arithmetic Mean-Geometric Mean Inequality
- Cauchy-Schwarz 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
- Schur's Inequality
- Triangle Inequality
- Trigonometric inequalities
- Trivial inequality
Problem solving tactics
substitution, telescoping, induction, etc. (write me please!)
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.
