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V prvem delu magistrskega dela je obravnavana konveksnost funkcij ene spremenljivke. Definiciji konveksnosti in osnovnim lastnostim le-te, sledi uvedba pomembnega pojma Jensenove neenakosti. Obravnavana je zveznost in odvedljivost konveksnih funkcij ter dokazana povezava med konveksnostjo in drugim odvodom neke funkcije. Sledi obravnava polkonveksnosti in kvazikonveksnosti, za zaključek prvega dela pa je predstavljena še funkcija Γ v povezavi s konveksnostjo. V drugem delu je obravnavana konveksnost funkcij več spremenljivk, pri čemer je poudarek na povezavi med konveksno množico in konveksno funkcijo ter med lastnostmi obeh. Zadnje poglavje se znova ukvarja z Jensenovo neenakostjo in z izpeljavo drugih neenakosti. In the first part of this master’s thesis, a convexity of functions of one variable is discussed. Following the definition and basic facts of convexity, the concept of Jensen’s inequality is introduced, followed by the continuity and differentiability of convex functions, where also a connection between the convexity and a second derivative of the function is proved. Then a concept of semi-convexity and quasiconvexity is introduced. The first part is concluded by presentation of the Γ function in connection with convexity. In the second part of thesis, the convexity of functions of more variables, or better, the convexity in a finite dimensional vector space is discussed. The emphasis is on the connection between convex sets and convex functions, where the properties of both are connected. The last section deals again with Jensen’s inequality and the other inequalities, derived from it. |
