K-Riemannov integral

Ovaj se diplomski rad bavi definiranjem \(\mathbb{K}\)-Riemannovog integrala. Kako bismo lakše definirali pojam \(\mathbb{K}\)-Riemannovog integrala u radu smo ponovili važna svojstva i definicije vezane za Riemannov integral kao što su: gornja i donja Darbouxova suma, infimum, supremum, gornje i donje međe, integrabilnost funkcije, monotonost funkcije i ograničenost funkcije. Promatrajući integral na polju \(\mathbb{K}\) kao potpolju od \(\mathbb{R}\) konstrukcija \(\mathbb{K}\)-Riemannovog integrala slična je konstrukciji Riemannovog integrala. U radu je također defininan pojam radijalne \(\mathbb{K}\)-derivacije te smo saznali postoji li poveznica između radijalne \(\mathbb{K}\)-derivacije i \(\mathbb{K}\)-Riemannovog integrala. Na kraju rada smo proširili klasične Hermite-Hadamardove nejednakosti na slučaj kada se uobičajeni Riemannov integral zamijeni s \(\mathbb{K}\)-Riemannovim integralom i pojam konveksnosti zamijeni s \(\mathbb{K}\)-konveksnosti.This master thesis introduces definition of \(\mathbb{K}\)-Riemann integral. In the beginning, we repeat some important properties and definitions regarding Riemann integral: upper and lower Darboux sum, infimum, supremum, upper and lower bound, function integrability, monotone function. Looking at integral on \(\mathbb{K}\) field as subfield of \(\mathbb{R}\), construction of \(\mathbb{K}\)-Riemann integral is similar to Riemann integral construction. In this thesis we also define radial \(\mathbb{K}\)-derivation and we find out a connection between radial \(\mathbb{K}\)-derivation and \(\mathbb{K}\)-Riemann integral. At the end of this thesis we extended classical Hermite-Hadamard inequalities to the case when the classical Riemann integral is replaced by \(\mathbb{K}\)-Riemann integral and the notion of convexity is replaced by \(\mathbb{K}\)-convexity