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 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 ( 1 , 2 )

of two numbers x1 and x 2 can be defined via the equation

− = − AM x1 x 2 AM

The 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

(1, 2, … , ) ≥1 (1,2,…,)

(0,2)=(1,0)=0

(1, 1)=1 and for positive distinct and numbers 1 2

− 2 1 (1,2)= 2−1 The following are some basic properties of the logarithmic means:

Logarithmic mean can be thought of as , the mean-value of the over = the interval . , The logarithmic mean can also be interpreted as the area under an exponential curve.

Since − = = − We also have the identity

, = Using this representation it is easy to show that , = (, ) We have the identity (, ) = (, ) (, ) which follows easily:

(, ) − − + = = (, ) 2( − ) − 2 The identric mean of two distinct positive real numbers is defined as: , 1 , = with

, = The slope of the joining the points and on the graph of the (, ) (, ) function

f(x) = x ln(x) is the natural of . (, ) Theorem 1. Suppose is a function with a strictly increasing .: , → Then

− − − 2 − − () + − for all in . , Let be defined by the equation − () = Then, − − − − 2 − − () + − is the sharpest form of the above inequality. Proof. By the , for all in , we have , ,

− () ′() − for some between and . Assuming without loss of generality by the assumption of the theorem we have ,

− − ′() Integrating both sides with respect to , we have − + − − − − and the inequality of the theorem follows. Let us now put

= − + − − − − Note that

= − − − () Moreover, since

= = − + − there exists an in such that . (, ) = 0 Since is strictly increasing, we have ′ = 0 for and = 0 for

Thus, is a minimum of and for () () all in , . Let us now assume that 0 .

Let us let The condition of the Theorem = . 1 is satisfied.

We compute / = And the inequality of the theorem becomes

(, ) (, )

Now, let us let . The condition of = Theorem 1 is satisfied. One can easily compute

=

And the inequality of the theorem becomes

(, ) (, ) Now let . Again the condition of = − Theorem 1 is satisfied. The of the theorem can be computed from the equation

= the logarithmic mean of a and b. The inequality of the theorem becomes

(, ) (, ) Finally, let us put . Again the = condition of Theorem 1 is satisfied.

In this case,

= The identric mean of a and b. The inequality of the theorem becomes

(, ) (, ) Thus, we now have for

(, ) (, ) (, ) (, ) (, ) ® Alzer H. 1985a. Über einen Wert, der zwischen dem geometrischen und dem artihmetischen Mittel zweier Zahlen liegt. In Elemente Math., 40 (1985), p. 22-24.

® Alzer H. 1985b. Ungleichungen für . In Elemente Math., 40 (1985), p. 120-123. ⁄ ⁄

® Beckenbach E. F. and R. Bellman. 1971. Inequalities. Third Edition. Ergebnisse der Mathematik, Band 30, Berlin and New York: Springer- Verlag.

® Carlson, B. C. 1965. A hypergeometric mean value. In Proc. Am. Math. Soc. 16 (1965), p. 759-766.

® Carlson B. C. and M. D. Tobey. 1968. A property of the hypergeometric mean value. In Proc. Am. Math. Soc. 19 (1968), p. 255-262.