is a monotonic function

           

            Here is a simple method to show the result,

using the generalized arithmetic mean > geometric mean of the form:

 

                   

where not all the a’s are equal.

 

 

 

                  

                  

                  

 

 

           Here you have the product of (n+1) numbers. They are not all equal since :

               

       

        Apply A.M.>G.M. for (n+1) numbers.

 

 

 

        Simplify.

 

 

        Bingo!