Inequalities:

A.M. ≥ G. M. ≥ H. M.:

Suppose a₁, a₂,... . . . ,  a_n be   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_n  be  n positive  real numbers  and m₁,  m₂, . .. . . . ,  m_n will 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^th Power:

Suppose a₁, a₂,......, a_n be 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      ( A.M. ≥ G. M. )

                                    => x+1/x ≥ 2.

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

                                    and 

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