Univariate inequalities based on Sobolev representations.

Here we derive very general univariate tight integral inequalities of Chebyshev-Grüss, Ostrowski types, for comparison of integral means and Information theory. These are based on well-known Sobolev integral representations of a function. Our inequalities engage ordinary and weak derivatives of the involved functions. We give also applications. On the way to prove our main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. Our results expand to all possible directions. [ABSTRACT FROM AUTHOR]