Half-linear dynamic equations and investigating weighted Hardy and Copson inequalities

S H Saker,Ghada Alnemer,A G Sayed,M Zakarya

doi:10.1186/s13662-020-03006-z

Abstract

In this paper, we employ some algebraic equations due to Hardy and Littlewood to establish some conditions on weights in dynamic inequalities of Hardy and Copson type. For illustrations, we derive some dynamic inequalities of Wirtinger, Copson and Hardy types and formulate the classical integral and discrete inequalities with sharp constants as particular cases. The results improve some results obtained in the literature.

Highlights

One of the applications of Hardy-type inequalities in dynamic equations was demonstrated in [19]
Hardy in [12] proved the discrete inequality ∞ s a(i) q ≤ q q∞aq(s), for q > 1, s ≥ 1, (1)s q–1 s=1 i=1 s=1 where a(s) is a positive sequence for s ≥ 1
Α and we prove that the conditions on the weights reduces to the solvability of the dynamic equation γ (s) –u (s) q–1 – ω(s) uσ (s) q–1 = 0, s ∈ [α, β]T, (14)

Summary

Introduction

One of the applications of Hardy-type inequalities in dynamic equations was demonstrated in [19]. Proof From the definition of v and by using the rules of derivative on time scales (15), we have, for t ≥ α, v = γ S θ–1 Sσ 1–θ + γ S θ–1 S1–θ .

Results

Conclusion