Some Complements of Cauchy’s Inequality on Time Scales

To unify the theory of continuous and discrete dynamic systems, in 1990 Hilger [16] proposed the study of dynamic systems on a time scale and developed the calculus for functions on a time scale (i.e., any closed subset of reals). The purpose of this paper is to establish some complements of Cauchy’s inequality on time scales, which extend some results of Cargo [6], Diaz, Goldman, and Metcalf [7–11], Goldman [12], Greub and Rheinboldt [13], Kantorovich [17], Schweitzer [29], Pólya and Szegö [26], and so forth. For other related results, we refer to [2, 3, 14, 15, 18–20, 23–27, 30, 31]. To do this, we briefly introduce the time scale calculus as follows.