On Hilbert's inequality on time scales

Saker Saker,H.M Rezk,A.A El-Deeb,Ravi Agarwal

doi:10.2298/aadm170428001s

Abstract

In this paper, we will prove some new dynamic inequalities of Hilbert's type on time scales. Our results as special cases extend some obtained dynamic inequalities on time scales.and also contain some integral and discrete in- equalities as special cases. We prove our main results by using some algebraic inequalities, H?older's inequality, Jensen's inequality and a simple consequence of Keller's chain rule on time scales.

Highlights

The original integral Hilbert’s inequality is given by (1)∞ ∞ f (x)g(y) dxdy ≤ π ∞f 2(x)dx g2(x)dx,0 0 x+y where f (x), g(x) are nonnegative functions and satisfy f 2(x)dx < ∞, and g2(x)dx < ∞.The constant π is the best possible
Before we present our main result, let us recall essentials about time scales
The Holder’s inequality, on time scales is given by b b γ b ν

Summary

Introduction

The original integral Hilbert’s inequality is given by (1)∞ ∞ f (x)g(y) dxdy ≤ π ∞f 2(x)dx g2(x)dx ,0 0 x+y where f (x), g(x) are nonnegative functions and satisfy f 2(x)dx < ∞, and g2(x)dx < ∞.The constant π is the best possible (see [8]). The Holder’s inequality, (see [3, Theorem 6.13]) on time scales is given by b b γ b ν Applying Holder’s inequality (11) on the right hand side of (25) with indices p and q, we have σ(s) a(η)Ah−1(σ(η))∆η

Results

Conclusion