The diamond integral reverse Hölder inequality and related results on time scales

Abstract

In this paper, we establish reverse Hölder's inequality on time scales via diamond integral, which is defined as an 'approximate' symmetric integral on time scales. Moreover, we give some generalizations of diamond integral Hölder's inequality which is due to Brito da Cruz et al. Several other related inequalities are also presented. MSC: 26D15; 26E70

Generalizations and refinements of three-tuple Diamond-Alpha integral Hölder’s inequality on time scales

“…Remark 4.10 For the inequality of Theorem 3.7 in Ref. [17], we generalize it in this paper and obtain the generalized inequalities in Theorem 4.8 and Theorem 4.9.…”

“…[17]) Let f , g, h :T → R be ♦-integrable on [ξ , σ ] T , p > 1 with q = p/(p -1). Then we have σ ξ h(δ) f (δ)g(δ) ♦δ ≤ [17]) Let f , g, h : T → R be ♦-integrable on [ξ , σ ] T , p > 1 with q = p/(p -1).…”

“…Aiming at the diamond-α integral Minkowski's inequality proposed by Theorem 3.5 in Ref [17],. we generalize it in this paper and obtain the three-tuple and n-tuple diamond-α inequalities (34)-(37).…”

“…Remark 3.16 For the inequality of Theorem 5.1 in the Reference[17], we put forward Theorem 3.14 and Theorem 3.15 as the generalization results.…”