Hijerarhija konveksnih funkcija

master's thesis

• Author: Elma Đaferović
• Mentor: Sanja Varošanec (mentor)
• Committee member: Sanja Varošanec (predsjednik povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Nela Bosner (član povjerenstva)
• Committee member: Marko Erceg (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Konveksne funkcije predstavljaju važno mjesto u matematici, jer imaju mnogobrojnu primjenu u različitim područjima matematike. Početak proučavanja konveksnih funkcija vezan je uz danskog matematičara J.L.W.V. Jensena. U ovom radu razmotrili smo pitanje hijerarhije klasa konveksnih funkcija, odnosno pokazali smo da vrijedi \[K(b)\subset S^{*}(b)\subset S(b).\] U prvom poglavlju ovog rada izrekli smo osnovne definicije i svojstva konveksnih funkcija. Također smo iskazali i dokazali Jensenovu nejednakost i Hermite-Hadamardovu nejednakost za konveksne funkcije. Nakon konveksnih funkcija u drugom i trećem poglavlju definirali smo zvjezdaste i superaditivne funkcije. Nakon što smo dokazali nekoliko teorema i propozicija vezane uz konveksne, zvjezdaste i superaditivne funkcije na intervalu \([0,b]\), poglavlje smo zaključili proučavajući hijerarhiju konveksnih funkcija. U četvrtom poglavlju proučavali smo integralne sredine, tj. konveksne funkcije u srednjem, zvjezdaste u srednjem i superaditivne funkcije u srednjem te razmotrili hijerarhiju danih klasa funkcija: \[K(b)\subseteq MK(b)\subseteq S^{*}(b) \subseteq S(b)\subseteq MS^{*}(b)\subseteq MS(b).\] U posljednjem poglavlju ovog rada posvetili smo se poopćenju konveksnih funkcija. Definirali smo \(m\)-konveksne, \(m\)-zvjezdaste, \(m\)-superaditivne, \(m\)-konveksne funkcije u Jensenovom smislu, slabe \(m\)-konveksne funkcije u Jensenovom smislu, \(m\)-superaditivne funkcije u Jensenovom smislu, slabe \(m\)-superaditivne funkcije te razmotrili pitanje hijerarhije klasa tih funkcija.

Abstract (english)

Convex functions play an important role in many areas of mathematics, because they have many applications in different fields of mathematics. Danish mathematician J.L.W.V Jensen is considered for the creator of this field of mathematics. In this paper we have studied the question of hierarchy of the convexity of functions. We proved strict inclusion: \[K(b)\subset S^{*}(b)\subset S(b).\] In the first section of this paper, we have given the several definitions and properties of convexity of functions. We have also expressed and proved Jensen’s inequality and Hermite-Hadamard’s inequality for convex functions. In Chapter 2 and 3 we defined starshaped and superadditive functions. After that we proved several theorems and propositions for convex, starshaped, and superadditive functions on \([0,b]\), we concluded chapter by studying hierarchy of convexity of this functions. In the fourth chapter, we extended our results by the arithmetic integral mean, respectively we studied the sets of functions which are convex, starshaped, respectively superadditive in the mean. We proved inclusions: \[K(b)\subseteq MK(b)\subseteq S^{*}(b) \subseteq S(b)\subseteq MS^{*}(b)\subseteq MS(b).\] At the end of this paper, we defined \(m\)-convex, \(m\)-starshaped functions, \(m\)-superadditive functions, Jensen \(m\)-convex functions, weak Jensen \(m\)-convex functions, Jensen \(m\)-superadditive functions, and weak \(m\)-superadditive functions. Some inclusions between such classes of functions are established.

• Keywords: konveksne funkcije; Jensen; hijerarhija klasa konveksnih funkcija; Hermite-Hadamardova nejednakost za konveksne funkcije
• Keywords (english): convex functions; Jensen; hierarchy of the convexity of functions; Hermite-Hadamard’s inequality for convex functions
• Language: croatian
• URN:NBN: urn:nbn:hr:217:166087
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2018-11-23 10:46:01