# Kvaasi-aritmeetilised keskmised ja Lagrange'i keskmised

## Date

2013

## Authors

## Journal Title

## Journal ISSN

## Volume Title

## Publisher

Tartu Ülikool

## Abstract

In this thesis, a continuous function is called a proper mean if it is symmetric, reflexive
and monotonic. We mainly observe quasi-arithmetic and Lagrangian means.
Lagrangian means are de ned as means obtained from applying the classical mean
value formula to a continuous and strictly monotonic function. Quasi-arithmetic
means are transformations of the common arithmetic mean. We de ne the quasiarithmetic
mean Qf associated with f : I ! I as
Qf (x; y) := f-1
f (x) + f (y)
2
x; y 2 I
and Lagrangian mean Lf as
Lf (x; y) :=
8<
:
f1
1
yx
Ry
x
f(t)dt
; if x 6= y;
x; if x = y:
Many well-known means like arithmetic and geometric mean are both Lagrangian
and quasi-arithmetic. The harmonic mean however is not Lagrangian but it is
quasi-arithmetic. It is shown that there is a close relationship between Lagrangian
and quasi-arithmetic means. This relationship can be made explicit through a
homeomorphism.
The purpose of this thesis is to give answers for two questions:
1. Which means can be represented according to some function f as quasiarithmetic
and Lagrangian mean?
2. How to describe function classes with the same quasi-arithmetic and Lagrangian
mean?
The answers are presented in the second and third chapter.
41
In the rst chapter, some of the well-known means are discussed. The de nition
of a proper mean is given and illustrated with some examples.
The second chapter of the thesis, is dedicated to quasi-arithmetic means. It is
shown that Qf = Qg if and only if g = f + where ; 2 R and 6= 0: A proof
of the famous Acz el theorem is given.
In the last chapter of the thesis, Lagrangian means are observed. It is shown that
Lf = Lg if and only if g = f + where ; 2 R and 6= 0.