chapter  1
54 Pages

Discrete Inequalities

Some of the most important inequalities for real or complex numbers and sequences in analysis are presented in this chapter. These include the Abel inequality; the Cauchy-Bunyakovsky-Schwarz (CBS) inequality; De Bruijn’s inequality; Cˇebysˇev’s inequality for synchronous sequences; the Biernacki, Pidek, and Ryll-Nardzewski inequality and its weighted form due to AndricaBadea refining the Gru¨ss inequality; and the Daykin-Eliezer-Carlitz inequality, which together with Wagner’s inequality provide interesting and different generalisations for the CBS inequality. Classical reverses for the CBS inequality due to Po´lya-Szego¨ and Cassels are also provided. Some fresh insight on the celebrated Ho¨lder and Minkowski inequalities for sequences of real numbers are presented, and various inequalities for convex functions, including those of Jensen, Slater, and Petrovic´, are investigated.