Kvaziaritmetičke sredine

master's thesis

• Author: Sanja Lovrić
• Mentor: Zlatko Pavić (mentor)
• Mentor: Sanja Varošanec (mentor)
• Committee member: Zlatko Pavić (predsjednik povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Committee member: Dragutin Svrtan (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2016-04-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom diplomskom radu proučavaju se kvaziaritmetičke sredine. Rad se sastoji od četiriju poglavlja. Prvo poglavlje se bavi pojmom konveksnosti. Dokazana su brojna svojstva konveksnih funkcija te navedeni primjeri istih. Drugo poglavlje izučava diskretni i integralni oblik Jensenove i Hermite-Hadamardove nejednakosti. Navedene nejednakosti korištene su pri uspoređivanju matematičkih sredina. U trećem poglavlju proučavaju se aritmetička, geometrijska i harmonijska sredina i osnovne integralne sredine: aritmetička, identrička i logaritamska. Na kraju poglavlja navedene sredine su poredane po veličini. Četvrto poglavlje se bavi diskretnim i integralnim kvaziaritmetičkim sredinama. Dokazan je osnovni teorem koji govori o poretku kvaziaritmetičkih sredina. Na kraju rada uvedene su potencijske sredine kao poseban slučaj diskretnih i logaritamske sredine kao poseban slučaj integralnih kvaziaritmetičkih sredina.

Abstract (english)

This graduate thesis studies quasiarithmetic means. The paper consists of four chapters. The first chapter deals with the concept of convexity. Numerous properties of convex functions are proven and examples of convex functions are provided. The second chapter studies discrete and integral form of Jensen’s and Hermite-Hadamard’s inequalities. These inequalities are used for comparison of mathematical means. The third chapter examines arithmetic, geometric and harmonic means and basic integral means: arithmetic, identric and logarithmic. At the end of the chapter all aforementioned means are sorted by size. The fourth chapter deals with discrete and integral quasiarithmetic means. The basic theorem which refers to order of quasiarithmetic means is proven. End of the paper introduces power means as a special case of discrete and logarithmic means as a special case of integral quasiarithmetic means.

• Keywords: kvaziaritmetičke sredine; konveksnost; Jensenova nejednakost; Hermite-Hadamardova nejednakost; potencijske sredine; logaritamske sredine
• Keywords (english): quasiarithmetic means; convexity; Jensen’s inequality; Hermite-Hadamard’s inequality; power means; logarithmic means
• Language: croatian
• URN:NBN: urn:nbn:hr:217:729607
• 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: 2019-02-28 12:06:09