In this paper, we use an identity of Fink and present some interesting identities and inequalities for real valued functions and r-convex functions respectively. We also obtain generalizations of some Hardy-Littlewood-Pólya type inequalities. In addition, we use the Čebyšev functional and the Grüss type inequalities and find the bounds for the remainder in the obtained identities. Finally, we present an interesting result related to the Ostrowski type inequalities