The main aim of this paper is to give improvements of various forms of the HermiteHadamard inequality, namely, that of Fejer, Lupas, Brenner-Alzer, Beesack-Pecaric. It is interesting that these improvements also imply the Hammer-Bullen inequality which deals with a comparison of the left-hand and the right-hand side of the Hermite-Hadamard inequality. These improvements are given in terms of positive linear functionals. Obtained results are used in constructing a new family of exponentially convex functions. Mathematics subject classification (2010): 26D15.