Visions of Mathematics: Glossary

1. Trivial Inequality:
If x is any real number, we have: $x^2 \geq 0$
This seems 'trivial' but is the basis for every other inequality

2. Mean Inequality
Let $a_{1}, a_{2}.. a_{n}$ be n positive numbers. Then $A\geq G\geq H$ where

$A = \frac{a_{1} +a_{2} + a_{3} +.. + a_{n} }{n}$

$G = \sqrt[n]{^{a_{1}*a_{2}*a_{3}..a_{n} }}$

$H = \frac{n}{\frac{1}{a_{1}} + \frac{1}{a_{2}} +..+ \frac{1}{a_{n}}}$

Sophie inequality

$x^2 + y^2 + xy \geq (x+y) ^2$

$x^2 + y^2 - xy \geq (x+y) ^2$

3. Triangle Inequality

4. Weistras's Inequality

5. Cauchy-Schwarz Inequality

6. Tchebycheff's Inequality

7. Holders Inequality

8. Ptolemy's Inequality

9. The Parellelogram Inequality

11. Toricelli's (or Fermat's) point for maxima/minima

12. Leibniz's Theorem

13. The Erodos-Modell Inequality

14. Jensen's Inequality

Glossary - Inequalities