Exponential-geometric mean

Abstract: A new concept of exponential-geometric mean is introduced and its properties are analyzed. The concepts and properties of means of a set of numbers are well studied in calculus. In [1], a mean μ of a set of numbers nixi,...,2,1, is defined as the value that satisfies the condition: nn xxxxxx,...,,max,...,,min 2121 . In this short note I introduce a new so-called exponential-geometric mean and give some of its properties. Definition: A lower (upper) exponential-geometric mean μ of two positive numbers a and b is: ba baba ab baba 10 . We can see that the exponential-geometric mean μ of any two positive numbers a and b conforms to the general definition of the mean. Indeed, without loss of generality, let’s assume ba , then: bb aa baabbaba ab ba ba bbaabbaa aa ba babababa ba ba ba bbaabbaa. Note that the other two combinations ba aaba and ba bbba may not suit the general definition of the mean. Examples: The lower and upper exponential-geometric means of numbers 1) 2 and 3 are 35.23232 23 and 55.23232 32