On some inequalities of the Grüss-Barnes and Borell type (Q1340553)

Proofs and generalizations of Barnes' [\textit{D. C. Barnes}, J. Math. Anal. Appl. 26, 82-87 (1969; Zbl 0186.37302)] and Borell's [\textit{C. Borell}, Uppsala Univ. Math. Res. Rep. No. 44 (1972); J. Math. Anal. Appl. 41, 300-312 (1973; Zbl 0249.26015)] inequalities are offered. The main tools are Chebyshev, Hardy-Littlewood-Pólya [\textit{G. H. Hardy}, \textit{J. E. Littlewood} and \textit{G. Pólya}: ``Inequalities'', reprint of the 2nd ed. (1988; Zbl 0634.26008), inequality No. 378 (first ed. 1934; Zbl 0010.10703)] and Lorentz [\textit{G. G. Lorentz}, Am. Math. Mon. 60, 176-179 (1953; Zbl 0050.28201)] convex functions and decreasing rearrangement.