Title (M,N) - konveksne funkcije pridružene paru sredina M i N Author Matija Ilić Mentor Sanja Varošanec (mentor)Committee member Sanja Varošanec (predsjednik povjerenstva)Committee member Igor Pažanin (član povjerenstva)Committee member Matija Kazalicki (član povjerenstva)Committee member Maja Starčević (član povjerenstva)Granter University of Zagreb Defense date and country 2015-07-10, Croatia Scientific / art field, NATURAL SCIENCES Abstract Gledano kroz povijest matematike, područje \((M,N)\) - konveksnih funkcija je relativno mlado. Prvi radovi na temu konveksnih funkcija pojavili su se krajem 19. stoljeća, a začetnikom tog područja matematike smatramo danskog matematičara J.L.W.V. Jensena. \((M,N)\) - konveksne funkcije počele su se proučavati tek u 20. stoljeću, a danas je općeprihvaćena sljedeća definicija. Neka je \(f:I\rightarrow \mathbb{R}^+\) neprekidna, gdje je \(I\subseteq \mathbb{R}^+\) i neka su \(M\) i \(N\) dvije funkcije sredina. Kažemo da je funkcija \(f\) \(\left(M,N\right)\)-konveksna ako za sve \(x,y\in I\) vrijedi: \[f\left(M\left(x,y\right)\right)\leq N\left(f\left(x\right),f\left(y\right)\right).\] U prvom poglavlju ovog rada obradili smo pojam konveksnih funkcija te dali nekoliko načina definiranja istih. Također smo iskazali i dokazali neke bitne nam teoreme za glavni dio rada poput Hermite-Hadamardova teorema. Poglavlje smo zaključili navođenjem svojstava konveksnih funkcija koje ćemo kasnije koristiti. Nakon konveksnih funkcija u drugom poglavlju prisjetili smo se poznate nejednakosti između harmonijske, geometrijske, aritmetičke i kvadratne sredine. U nastavku smo definirali težinske sredine i težinske sredine reda t. Za kraj drugog poglavlja smo iskazali i dokazali tzv. teorem o monotonosti. U glavnom dijelu ovog rada, 3. i 4. poglavlju, u potpunosti smo se posvetili proučavanju \((M,N)\) - konveksnih funkcija. Nakon same definicije, proučavali smo kriterije i svojstva konveksnosti funkcija pridruženih paru sredina, gdje smo za sredine uzimali harmonijsku, geometrijsku i aritmetičku sredinu. Nakon toga smo dokazali nekoliko teorema i propozicija \((M,N)\) - konveksnih funkcija vezanih uz kvazi-aritmetičke sredine i sredine reda t. U posljednjem poglavlju ovog rada proučavali smo \((A,G)\) i \((G,G)\) - konveksne funkcije. Ovdje smo definirali gama i beta funkciju, logaritamsku sredinu i dokazali razne teoreme vezane uz \((A,G)\) i \((G,G)\) - konveksne funkcije, često koristeći Hermite - Hadamardov teorem i infinitezimalni račun.
Abstract (english) Viewed through history of mathematics, area of \((M,N)\) - convex functions is relatively young. The first works on the theme of convex functions appeared in the late 19th century, and for the pioneer of this field of mathematics is considered Danish mathematician J.L.W.V. Jensen. \((M,N)\) - convex function have begun to be studied only in the 20th century, and today it is generally accepted the following definition. Let \(f:I\rightarrow \mathbb{R}^+\) be continuous, where \(I\) is a subinterval of \(\mathbb{R}^+\), and let \(M\) and \(N\) be any two mean functions. We say that \(f\) is \(\left(M,N\right)\)-convex if for all \(x,y\in I\): \[f\left(M\left(x,y\right)\right)\leq N\left(f\left(x\right),f\left(y\right)\right).\] In the first section of this paper we have process the concept of convex function and gave several ways of defining them. We have also expressed and proved some important theorems for our main part of the work, such as the Hermite-Hadamard theorem. We concluded chapter by stating the properties of convex functions which we will use later. In Chapter 2 we reminded ourselves of the known inequalities between the harmonic, geometric, arithmetic and square means. Below, we defined the weighted mean and the weighted mean of order t. At the end of the second chapter we expressed and proved the so-called Theorem of monotony. The main parts of this work are Chapters 3 and 4. In Chapter 3 we defined \((M,N)\) - convex functions and studied the criteria and properties of convexity of functions according to the pair of means, where for means we have taken harmonic, geometric and arithmetic mean. After that we proved several theorems and propositions for \((M,N)\) - convex functions related to the quasi-arithmetic means and the means of order t. The last chapter of this work is devoted to \((A,G)\) and \((G,A)\) - convex functions. We defined the gamma and beta functions, logarithmic mean and proved various theorems related to \((A,G)\) and \((G,A)\) - convex functions.
Keywords
konveksne funkcije
Jensen
Hermite-Hadamardov teorem
harmonijska sredina
geometrijska sredina
aritmetička sredina
kvadratna sredina
teorem o monotonosti
gama funkcija
beta funkcija
logaritamska sredina
Keywords (english)
convex functions
Jensen
Hermite-Hadamard theorem
harmonic mean
geometric mean
arithmetic mean
square mean
Theorem of monotony
gamma function
beta function
logarithmic mean
Language croatian URN:NBN urn:nbn:hr:217:516404 Study programme Title: Mathematics Education; specializations in: Mathematics Education Type of resource Text File origin Born digital Access conditions Open access Terms of use Created on 2019-01-30 11:48:28
