Ilhan M. Izmirli George Mason University [email protected] In this paper our goal is to prove the well-known chain of inequalities involving the harmonic, geometric, logarithmic, identric, and arithmetic means using nothing more than basic calculus. Of course, these results are all well-known and several proofs of them and their generalizations have been given: Hardy, Littlewood, and Polya (1964) Carlson (1965) Carlson and Tobey (1968) Beckenbach and Bellman (1971) Alzer (1985a, 1985b) Our purpose here is to present a unified approach and give the proofs as corollaries of one elementary theorem. = For a sequence of numbersx {x1, x2 ,..., xn}we will let
n
∑x j − AM (x , x ,..., x ) = AM (x) = j 1 1 2 n n
n = = /1 n GM (x1, x2 ,..., xn ) GM (x) ∏x j j=1 and
n = = HM (x1, x2 ,..., xn ) HM (x) n ∑ 1 j=1 x j to denote the well-knownarithmetic , geometric, and harmonic means , also called the Pythagorean means ( PM ). The Pythagorean means have the obvious properties:
1. PM(x1,x2,...,xn)is independent of order
2. PM(x,x,...,x)=x
= 3. PM(bx 1,bx 2,...,bx n) bPM(x1,x2,...,xn) 1.
2. is always a solution of a simple equation. In part icular, the arithmetic mean ( 1 , 2 )
of two numbers x1 and x 2 can be defined via the equation
− = − AM x1 x 2 AM
The harmonic mean satisfies the same relation with reciprocals, that is, it is a solutio n of the equation
1 − 1 = 1 − 1 x1 HM HM x 2
The geometric mean of two numbers x1 and x 2 can be visualized as the solution of the equation
x 1 = GM GM x 2 1. = ( )( )
2. 1 1 1, = 1 1 ( 1, ) 1
3. 1 1 1 2 ( 1 + 2 + ⋯ + ) + + ⋯ + ≥ 1 2
This follows because