An Elementary Proof of the Mean Inequalities

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.


Pythagorean Means
, , , n .The Pythagorean means have the obvious properties: , , , , PM x x is always a solution of a simple equa- tion.In particular, the arithmetic mean of two numbers 1 x and 2 x can be defined via the equation 1 2

AM x x AM   
The harmonic mean satisfies the same relation with reciprocals, that is, it is a solution of the equation The geometric mean of two numbers 1 x and 2 x can be visualized as the solution of the equation

Logarithmic and Identric Means
The logarithmic mean of two non-negative numbers x and 2 x is defined as follows: The following are some basic properties of the logarithmic means: 1) Logarithmic mean

 
, LM a b  can be thought of as the mean-value of the function 2) The logarithmic mean can also be interpreted as the area under an exponential curve. Since We also have the identity

LM x y
Using this representation it is easy to show that   LM cx cy cLM x y  1) We have the identity , , ,

LM x y x y x y  
To define the logarithmic mean of positive numbers The identric mean of two distinct positive real numbers 1 2 , x x is defined as: slope of the secant line joining the points The on the graph of the function It can be g s according by th

The Main Theorem
eneralized to more variable e mean value theorem for divided differences.
is the sharpest form of the above inequality.r all ,


Proof.By the Mean Value Theorem, fo s t in

 
,b , we have a for some u between s and t .Assu ing without loss m of generality , s t  by the assumption of the theorem we have Integrating both sides with respect to t we have and the inequality of the theorem follows.
Let us now put a b

Applications to Mean Inequalities
will extend the well-known chain of inequalities We to the more refined using nothing more than the mean value theorem of dif-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 he two numbers.ferential calculus.

of t
Let us now assume that 0 .
The condition of the Theorem 1 is satisfied.Solving the equation where H HM a b  Hence the left-hand side of the inequality becomes The condition of Theorem 1 is sfied.We can easily compute the 0 s sati of the theorem from the equation   [5] H. Alzer, "Übereinen Wert, der zwischendemgeometrischen und demartihmetischen Mittelzweier Zahlenliegt," Elemente der Mathematik, Vol. 40, 1985, pp. 22-24.elds    

For
well known arithmetic, geometric, and harmonic means, also called the Pythagorean means Again the condition of Theo- rem 1 is satisfied.The 0 s of the theorem can be com-