An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value.
a ≠ b says that a is not equal to b
a < b says that a is less than b
a > b says that a is greater than b
(those two are known as strict inequality)
a ≤ b means that a is less than or equal to b
a ≥ b means that a is greater than or equal to b.
CHAPTER 1. ARITHMETIC MEAN - GEOMETRIC MEAN (AM-GM) INEQUALITY
states that for any set of nonnegative real numbers, the arithmetic mean of the set is greater than or equal to the geometric mean of the set. Algebraically, this is expressed as follows.
An inequality between these means states that for any two positive numbers following inequality holds:
1.1. Theorem:
Proof:
Firstly, we’ll show that
For , we have
Equality holds if and only if , i.e. .
Furthermore, for we have
So with equality if and only if
i.e.
Finally, we will show that
We have
Equality holds if and only if i.e. .
1.2. Lemma:
If are positive numbers whose product is equal to 1 , then , with equality only when .
Proof :
By induction on . The case being trivial, we suppose . Let be positive numbers with . Without loss of generality, we can assume that and . Thus we have
i.e.,
Since are positive numbers with product equal to 1 , by the inductive hypothesis we have
Adding the inequalities (1) and (2), we get
If equality holds in (3) then we must have equality in (1), that is, or , where
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