Konveksne funkcije i njihova poopćenja u realnoj analizi

master's thesis

• Author: Viktor Vranar
• Mentor: Jadranka Mićić Hot (mentor)
• Mentor: Sanja Varošanec (mentor)
• Committee member: Jadranka Mićić Hot (predsjednik povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Željka Milin Šipuš (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Pojam i svojstva konveksnih funkcija relativno su mladi u području matematike. Tek u 19. stoljeću danski matematičar J.L.W.V. Jensen počeo je pisati radove koji sadrže konveksne funkcije kao zasebnu ideju. Jensena zato nazivamo začetnikom toga područja. Navedimo ovdje osnovnu definiciju konveksne funkcije. Neka je \(I \subseteq \mathbb{R}\) interval. Kažemo da je funkcija \(f:I\rightarrow \mathbb{R}\) konveksna ako za svaki izbor točaka \(x, y\in I\) i svaki \(t\in [0,1]\) vrijedi \[ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y). \] U prvom poglavlju uveli smo pojam konveksnih funkcija te jednu ekvivalentnu karakterizaciju. Zatim smo naveli nekoliko karakterizacija konveksnih funkcija te najvažniji rezultat vezan uz konveksne funkcije- Hermite-Hadamardovu nejednakost. Pokazali smo i da je njezina lijeva nejednakost jaca od desne. Zatim smo dokazali ekvivalentnost konveksnosti i Hermite-Hadamardove nejednakosti. Vidjeli smo da cak i samo lijeva ili samo desna strana Hermite-Hadamardove nejednakosti povlače konveksnost. U drugom poglavlju bavili smo se poopćenjima pojma konveksnosti funkcije. Najprije smo uveli pojam logaritamski konveksnih funkcija te oblik Hermite-Hadamardove nejednakosti za logaritamski konveksne funkcije. Zatim smo definirali i dokazali neka svojstva i Hermite-Hadamardov tip nejednakosti za log-konveksne funkcije. Slično smo nastavili redom za ova poopćenja konveksnih funkcija: eksponencijalno konveksne funkcije, superkvadratne funkcije, preinveksne funkcije, h-konveksne funkcije, r-konveksne funkcije te naposljetku jako konveksne funkcije.

Abstract (english)

Definition and properties of convex functions are relatively new in mathematics. Only in 19th century a Danish mathematican J.L.W.V. Jensen started doing research and publishing works that contained convex functions as a new concept. For that reason we call Jensen the originator of this area. Let us define convex functions. Let \(I \subseteq \mathbb{R}\) be an interval. We say that a function \(f:I\rightarrow \mathbb{R}\) is convex if for every \(x, y\in I\) and every \(t\in [0,1]\) holds inequality \[ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y). \]: In the first chapter we’ve introduced a concept of convex functions and one equivalent statement. Then we’ve introduced few characterizations of convex functions as well as the most important result in the area, which is the Hermite-Hadamard inequality. We have also showed that its left side is stronger than its right one. We have then proved equivalence of convex and the Hermite-Hadamard inequality. We have also seen that either left or right side implies convexity. In the second chapter we have been studying generalisations of convex functions. First we have introduced a term of logarithmically convex functions. Then we have defined and proved some properties and Hermite-Hadamard type inequality for log-convex functions. We have continued in similar order for: exponentially convex functions, superquadratic functions, preinvex functions, h-convex functions, r-convex functions and, at last, strongly convex functions.

• Keywords: konveksne funkcije; Jensen; Hermite-Hadamardova nejednakost
• Keywords (english): convex functions; Jensen; Hermite-Hadamard inequality
• Language: croatian
• URN:NBN: urn:nbn:hr:217:054349
• 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: 2019-07-05 11:51:43