Jako konveksne funkcije

master's thesis

• Title (english): Strongly convex functions
• Author: Lucija Novak
• Mentor: Sanja Varošanec (mentor)
• Committee member: Sanja Varošanec (predsjednik povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Tema ovoga rada su jako konveksne funkcije. Rad je podijeljen na pet poglavlja. Prvo poglavlje sadrži osnovne pojmove i teoreme koje koristimo kroz rad. Dana je definicija, primjeri i osnovna svojstva konveksnih funkcija, a zatim su opisane operacije s konveksnim funkcijama. Na kraju prvog poglavlja dana je veza konveksnih funkcija s Jensenovom nejednakosti i Hermite-Hadamardovom nejednakosti koja je ujedno i karakterizacija konveksne funkcije. U drugom poglavlju uvodimo pojam jako konveksnih funkcija kao poopćenje konveksnih funkcija. Osnovna definicija jako konveksne funkcije glasi: Neka je \(I\subset\mathbb{}{R}\) interval i \(c\) pozitivan broj. Znamo da je funkcija \(f:I \rightarrow \mathbb{R}\) jako konveksna funkcija s modulom \(c\) ako vrijedi \( f(\lambda x+(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y)-c\lambda(1-\lambda)(x-y)^2,\) za svaki \(x,y \in I\) i \(\lambda\in [0,1] \). Također, u drugom poglavlju poopćavaju se neka svojstva konveksnih funkcija za jako konveksne funkcije. U trećem poglavlju navodimo i dokazujemo poopćenje Jensenove i Hermite-Hadamardove nejednakosti za jako konveksne funkcije te nekoliko teorema povezanih s Hermite-Hadamardovom nejednakosti. Četvrto poglavlje daje uvid u karakterizaciju unitarnog prostora pomoću jako konveksnih funkcija. U petom poglavlju dajemo poopćenje teorema separacije za jako konveksne funkcije.

Abstract (english)

The topic of this paper is strongly convex functions. The paper is divided into five chapters. The first chapter contains basic concepts and theorems that we use in paper. The definition, examples and description of basic properties of convex functions are given, and then the operations with convex functions are described. At the end of the first chapter there is a connection of convex functions with Jensen’s inequality and the Hermite-Hadamard inequality, which is also the characterization of convex functions. In the second chapter, we introduce the notion of strongly convex functions as a generalization of convex functions. Definition of convex function is: Let \(I\subset\mathbb{}{R}\) be an interval and \(c\) be a positive number. A function \(f:I \rightarrow \mathbb{R}\) is called strongly convex function with modulus \(c\) if \( f(\lambda x+(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y)-c\lambda(1-\lambda)(x-y)^2,\), for all \(x,y \in I\) and \(\lambda\in [0,1] \). Also, in the second chapter, some properties of convex functions are generalized for strongly convex functions. In the third chapter, we state and prove the generalization of the Jensen and Hermite-Hadamard inequalities for strongly convex functions and several theorems related to the Hermite-Hadamard inequality. The fourth chapter provides an insight into the characterization of the unitary space using strongly convex functions. In the fifth chapter, we give a generalization of the separation theorem for strongly convex functions.

• Keywords: Jensenova nejednakost; Hermite-Hadamardova nejednakost; unitarni prostor; teorem separacije
• Keywords (english): Jensen’s inequality; Hermite-Hadamard inequality; unitary space; separation theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:345357
• Study programme: Title: Mathematics and Computer Science Education; specializations in: Mathematics and Computer Science Education; Course: Mathematics and Computer Science Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-11-03 11:42:23