Hilbert-type inequalities for time scale nabla calculus

H M Rezk,Kottakkaran Sooppy Nisar,Abdel-Haleem Abdel-Aty,M Zakarya,Ghada Alnemer,H A Abd El-Hamid

doi:10.1186/s13662-020-03079-w

H M Rezk, Kottakkaran Sooppy Nisar + Show 4 more

Open Access

https://doi.org/10.1186/s13662-020-03079-w

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Abstract

This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequality, and Jensen’s time scale inequality. This generalization allows us not only to unify all the related results that exist in the literature on an arbitrary time scale, but also to obtain new outcomes that are analytical to the results of the delta time scale calculation.

Highlights

In recent years, Hilbert’s dual-series inequality and its integral form [1, pp. 253–254] have been granted significant attention by many scholars
In [17] the researchers concluded some generalizations of inequalities (1) and (2) for time scale delta calculus
The main theorems are inspired from the paper [18] which presents the corresponding results for time scale delta calculus

Summary

Introduction

Hilbert’s dual-series inequality and its integral form [1, pp. 253–254] have been granted significant attention by many scholars (for example, see [2,3,4,5,6,7,8,9,10]). Hilbert’s dual-series inequality and its integral form [1, pp. 253–254] have been granted significant attention by many scholars (for example, see [2,3,4,5,6,7,8,9,10]). G. Pachpatte [11] established a new inequality close to that of Hilbert as follows. In the same article [11], Pachpatte demonstrated the integral version of (1) as follows.

Then pq

The set