Kvaasi-aritmeetilised keskmised ja Lagrange'i keskmised
| dc.contributor.advisor | Leiger, Toivo, juhendaja | |
| dc.contributor.author | Päll, Kärt | |
| dc.contributor.other | Tartu Ülikool. Matemaatika-informaatikateaduskond | et |
| dc.contributor.other | Tartu Ülikool. Matemaatika instituut | et |
| dc.date.accessioned | 2013-06-18T13:14:09Z | |
| dc.date.available | 2013-06-18T13:14:09Z | |
| dc.date.issued | 2013 | |
| dc.description.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. | en |
| dc.identifier.uri | http://hdl.handle.net/10062/31022 | |
| dc.language.iso | et | et |
| dc.publisher | Tartu Ülikool | et |
| dc.subject.other | Lagrange'i keskmine | et |
| dc.subject.other | kvaasi-aritmeetiline keskmine | et |
| dc.subject.other | Aczeli teoreem | et |
| dc.subject.other | bisümmeetria | et |
| dc.subject.other | bakalaureusetööd | et |
| dc.title | Kvaasi-aritmeetilised keskmised ja Lagrange'i keskmised | et |
| dc.type | Thesis | et |