On Generalized Means

Every student of elementary mathematics is acquainted with the concepts of the arithmetic mean, the harmonic mean, and the geometric mean of sets of numbers. These concepts are also common mathematical tools used in the analysis of data in many applied sciences. It may not be so well known that there exist various types of means of which those mentioned above are very special cases. It is the object of this paper to give postulational definitions of the concept of means in general, and to derive certain relations between different kinds of means. In this connection it will also be shown that certain classes of functions, namely convex functions, can be defined in terms of the notion of means