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In this paper we will extend the well-known chain of inequalities involving the Pythagorean means, namely the harmonic, geometric, and arithmetic means to the more refined chain of inequalities by including the logarithmic and identric 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. See [1-6] for more information. Our goal here is to present a unified approach and give the proofs as corollaries of one basic theorem.

For a sequence of numbers we will let

and

to denote the well known arithmetic, geometric, and harmonic means, also called the Pythagorean means.

The Pythagorean means have the obvious properties:

1) is independent of order 2)

3)

4) is always a solution of a simple equation. In particular, the arithmetic mean of two numbers and can be defined via the equation

The harmonic mean satisfies the same relation with reciprocals, that is, it is a solution of the equation

The geometric mean of two numbers and can be visualized as the solution of the equation

1)

2)

3)

This follows because

The logarithmic mean of two non-negative numbers and is defined as follows:

and for positive distinct numbers and

The following are some basic properties of the logarithmic means:

1) Logarithmic mean can be thought of as the mean-value of the function over the interval.

2) 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

1) We have the identity

which follows easily:

To define the logarithmic mean of positive numbers, we first recall the definition of divided differences for a function at points, denoted as

For

and for and,

We now define

So for example for n = 2, we get

The identric mean of two distinct positive real numbers is defined as:

with.

The slope of the secant line joining the points

and on the graph of the function

is the natural logarithm of.

It can be generalized to more variables according by the mean value theorem for divided differences.

Theorem 1. Suppose is a function with a strictly increasing derivative. Then

for all in.

Let be defined by the equation

Then,

is the sharpest form of the above inequality.

Proof. By the Mean Value Theorem, 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.

Since is strictly increasing, we have

for

and

for

Thus, is a minimum of and for all

We will extend the well-known chain of inequalities

to the more refined

using nothing more than the mean value theorem of differential calculus. All of these are strict inequalities unless, of course, the numbers are the same, in which case all means are equal to the common value of the two numbers.

Let us now assume that

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

we find

where

Hence the left-hand side of the inequality becomes

Thus we have

implying

or

Let us let. The condition of Theorem 1 is satisfied. We can easily compute the of the theorem from the equation

as

Our inequality becomes

Implying,

that is

Now let. Again the condition of Theorem 1 is satisfied. The of the theorem can be computed from the equation

as

where

Since

Thus,

where

Consequently our inequality becomes

implying

that is,

Finally, let us put. Again the condition of Theorem 1 is satisfied. Since in this case

the of the theorem can be computed as

The right-hand side of the inequality becomes

The integral on the left-hand side of our inequality yields

implying

or

Thus, we now have for