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.identifier.uri http://hdl.handle.net/10062/31022 dc.description.abstract In this thesis, a continuous function is called a proper mean if it is symmetric, reflexive en 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. 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
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