On Some Inequalities of the Grüss-Barnes and Borell Type

Some new generalizations of the Grüss-Barnes and Borell inequalities are proved. A new proof, using the classical Chebyshev inequality, for the case of two functions is presented and applied. A crucial inequality for n functions by Borell [J. Math. Anal. Appl.41 (1973), 300-312] is also discussed. Some recent results are sharpened and complemented. Interpolated or weighted versions of some of the inequalities are pointed out.