Inequalities:

A.M. ≥G. M. ≥H. M.:

假设一个₁, a₂,... . . . , a_nbe n positive real numbers, then it may be defined that A≥G≥H. Moreover equality holds at either place if and only if a₁= a₂= ....... = a_n.

Weighted Means:

Consider a₁, a₂, a₃. . . , a_nbe n positive real numbers and m₁, m₂, . .. . . . , m_nwill be n positive rational numbers. Then we may described weighted Arithmetic mean (A*), weighted Geometric mean (G*) and weighted harmonic mean (H*) as

It may be defined that A*≥G*≥H*. Moreover equality holds at either place if and only if a₁= a₂= . . . . . = a_n.

Arithmetic Mean of m^thPower:

假设一个₁, a₂,......, a_nbe n positive real numbers (not all equal) and let m be a natural number, then

if m ∈ R -[0, 1].

Yet if m ∈ (0, 1) , then

Clearly if m∈{0, 1} , then

Problem: Prove that x +1/x ≥2, if x > 0 and x + 1/x ≤-2 , if x < 0 .

Solution:Since(点≥G. M. )

=>x+1/x ≥2.

If x < 0 , let y = -x , then y > 0

and

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