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master's thesis
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Schur - convex Function
Martina Dorić (2017)
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Pure Mathematics
Probability and Mathematical Statistics Research Group
Metadata
TitleSchur - konveksne funkcije
AuthorMartina Dorić
Mentor(s)Mihaela Ribičić Penava (thesis advisor)
Abstract
Cilj ovog rada je sistematizirati neke rezultate o Schur-konveksnim funkcijama koji omogućuju njihovu identifikaciju te ukazati na njihovu ulogu u izvođenju nejednakosti. Prvo se definira majorizacija i navedena su dva motivacijska ekonomska primjera. To su Lorenzova krivulja koja omogućuje grafičku vizualizaciju pojma majorizacije i Daltonovi transferi koji služe kao motivacija definicije T-transformacija. Umnožak T-transformacija je dvostruko-stohastička matrica, a dvostruko stohastičkim matricama se može karakterizirati majorizacija. Schurkonveksne funkcije su funkcije koje čuvaju uredaj majorizacije. Simetričnost koja je važno svojstvo Schur-konveksnih funkcija proizlazi iz činjenice da je svaki vektor majoriziran sa svakom svojom permutacijom. Navedene su tvrdnje koje omogućavaju karakterizaciju i identifi- kaciju Schur-konveksnih funkcija, od kojih su najznačajnije karakterizacije diferencijabilnih Schur-konveksnih funkcija. Navedene su veze Schur-konveksnih funkcija s nekim klasama funkcija poput konveksnih i kvazikonveksnih funkcija te neki primjeri. Schur-konveksne funkcije se primjenjuju u mnogim područjima matematike, od kojih je navedeno nekoliko primjera iz geometrije trokuta i matrične teorije.
Keywordsmajorization Lorenz curve Dalton transfer T-transform doubly stohastic matrix Schur-convex function Schur-concave function convex function quasi-convex function
Parallel title (English)Schur - convex Function
Committee MembersDomagoj Ševerdija (committee chairperson)
Mihaela Ribičić Penava (committee member)
Ljerka Jukić Matić (committee member)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
Lower level organizational unitsChair of Pure Mathematics
Probability and Mathematical Statistics Research Group
PlaceOsijek
StateCroatia
Scientific field, discipline, subdisciplineNATURAL SCIENCES
Mathematics
Mathematical Analysis
Study programme typeuniversity
Study levelgraduate
Study programmeMathematics; specializations in: Financial and Statistical Mathematics, Mathematics and Computer Science, Industrial and Applied Mathematics
Study specializationFinancial and Statistical Mathematics
Academic title abbreviationmag.math.
Genremaster's thesis
Language Croatian
Defense date2017-10-20
Parallel abstract (English)
The purpose of this paper is to systematize some results on Schur convex functions that enable their identification and to point out their role in deriving inequalities. Firstly majorization is defined and two economics examples are given as motivation. Those are Lorenz curve that allows graphical visualisation of majorization and Dalton transfers that are used as motivation for definition of T-transforms. Products of T-transforms are doubly stohastic matrices and doubly stohastic matrices are used to characterize majorization. Schur-convex functions are functions that preserve the ordering of majorization. Symmetry is an important property of Schur-convex functions that arises from the fact that every vector is majorized by its permutations. Results are given that allow characterizations and identifi- cation of Schur-convex functions, where most relevant are characterizations of diferentiable Schur-convex functions. Schur-convex functions are linked to convex and quasi-convex functions and examples are given. Schur-convex functions have many mathematical applications, such as in geometry of triangles and matrix theory.
Parallel keywords (Croatian)majorizacija Lorenzova krivulja Daltonov transfer T-transformacija dvostruko-stohastiˇcka matrica Schur-konveksna funkcija Schur-konkavna funkcija konveksna funkcija kvazikonveksna funkcija
Resource typetext
Access conditionOpen access
Terms of usehttp://rightsstatements.org/vocab/InC/1.0/
URN:NBNhttps://urn.nsk.hr/urn:nbn:hr:126:731051