Generalized Inequalities of Hilbert-Type on Time Scales Nabla Calculus

In the time (1862–1943), David Hilbert proved Hilbert’s double series inequality without an exact determination of the constant in his lectures on integral equations. If $\left\{{\beta }_m\right\}$ and $\left\{d_n\right\}$ are two real sequences such that $0<{\sum }_{m=1}^{\infty }{\beta }_m^2<\infty$ and $0<{\sum }_{n=1}^{\infty }{d}_n^2<\infty ,$ then

In 1911, Schur [1] proved that $\pi$ in (1) is sharp and also discovered the integral analogue of (1), which became known as the Hilbert integral inequality in the form where $\mathrm{\Xi }$ and $\mathrm{{\rm Y}}$ are measurable functions such that, $0<{\int }_0^{\infty }{\mathrm{\Xi }}^2\left(\tau \right)d\tau <\infty$ and $0<{\int }_0^{\infty }{\mathrm{{\rm Y}}}^2\left(y\right)dy<\infty$.

In 1925, by introducing one pair of conjugate exponents $(p,$ $q)$ with $1/p+1/q=1,$ Hardy [2] gave an extension of (1) as follows. If $p,q>1,$ ${\beta }_m,d_n\ge 0$ such that $0<{\sum }_{m=1}^{\infty }{\beta }_m^p<\infty$ and $0<{\sum }_{n=1}^{\infty }{d}_n^q<\infty ,$ then where the constant $\pi /sin(\pi /p)$ in (3) is sharp. In 1934, Hardy et al. [3] proved the equivalent integral analog of (3) in the form where $\mathrm{\Xi }$ and $\mathrm{{\rm Y}}$ are measurable non-negative functions such that $0<{\int }_0^{\infty }{\mathrm{\Xi }}^p\left(\tau \right)d\tau <\infty$ and $0<{\int }_0^{\infty }{\mathrm{{\rm Y}}}^q\left(y\right)dy<\infty$.

In 1998, Pachpatte [4] gave a new inequality closed to that of Hilbert as follows: Let $\beta \left(\iota \right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $d\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $\beta \left(0\right)=d\left(0\right)=0.$ Then where $\nabla {\beta }_{\iota }={\beta }_{\iota }-{\beta }_{\iota -1},$ $\nabla d_{\vartheta }=d_{\vartheta }-d_{\vartheta -1}$ and

In 2000, Pachpatte [5] generalized (5) by introducing one pair of conjugate exponents $(\lambda ,\mu )$, such that $\lambda ,\mu >1$ with $1/\lambda +1/\mu =1$. Then, it is established that if $\beta \left(\iota \right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $d\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $\beta \left(0\right)=d\left(0\right)=0,$ then where $\nabla {\beta }_{\iota }={\beta }_{\iota }-{\beta }_{\iota -1},$ $\nabla d_{\vartheta }=d_{\vartheta }-d_{\vartheta -1}$ and

In 2002, Kim et al. [6] generalized (6) and proved that if $\lambda ,$ $\mu >1,$$\beta \left(\iota \right):\left\{0,1,2,\dots ,p\right\}\subset \mathbb{N}\to \mathbb{R}$ and $d\left(\vartheta \right):\left\{0,1,2,\dots ,q\right\}\subset \mathbb{N}\to \mathbb{R}$ with $\beta \left(0\right)=d\left(0\right)=0,$ then where $\nabla {\beta }_{\iota }={\beta }_{\iota }-{\beta }_{\iota -1},$ $\nabla d_{\vartheta }=d_{\vartheta }-d_{\vartheta -1}$ and

Furthermore, the researchers [6] proved the continuous analog of (7) and proved that if $\lambda ,$ $\mu >1,$ and $\mathrm{\Xi }\left(\iota \right),$ $\mathrm{{\rm Y}}\left(t\right)$ are real continuous functions on the intervals $\left(0,\tau \right),$ $\left(0,y\right),$ respectively, and let $\mathrm{\Xi }\left(0\right)=\mathrm{{\rm Y}}\left(0\right)=0.$ Then for $\tau ,$$y\in \left(0,\infty \right)$, where

In recent decades, a new theory has been discovered to unify the continuous calculus and discrete calculus. Many authors proved some dynamic inequalities of Hilbert type, its generalizations and also the reversed forms, see the papers [7,8,9].

The goal of this article is to use time scales nabla calculus to prove various dynamic inequalities for Hilbert-type. Furthermore, we establish some generalized inequalities of Hilbert type by using submuliplicative and bounded functions.

The organization of the paper as follows. In Section 2, we show some basics of the time scale theory and some lemmas needed in Section 3 where we prove our results. Our results as special cases give the inequalities (7) and (8) proved by Kim et al. [6].

For a time scale $\mathbb{T}$, we define the backward jump operator as following $\rho \left(\tau \right):=sup\{\iota \in \mathbb{T}:\iota <\tau \}.$ Let $\mathrm{\Xi }:\mathbb{T}\to \mathbb{R}$ be a function, we say that $\mathrm{\Xi }$ is ld-continuous if it is continuous at each left dense point in $\mathbb{T}$ and the right limit exists as a finite number for all right dense points $t\in \mathbb{T}.$ The set of all such ld–continuous functions is ushered by $C_{ld}(\mathbb{T},\mathbb{R})$ and for any function $\mathrm{\Xi }:\mathbb{T}\to \mathbb{R}$, the notation ${\mathrm{\Xi }}^{\rho }\left(\tau \right)$ denotes $\mathrm{\Xi }\left(\rho \right(\tau \left)\right).$ For more information about the time scale calculus, see [10,11].

The nabla derivative of $uv$ and $u/v$ (where $v\left(\tau \right)v^{\rho }\left(\tau \right)\ne 0$) are and

Throughout the article, we will assume that the functions are ld-continuous functions on ${[\beta ,d]}_{\mathbb{T}}:=[\beta ,d]\cap \mathbb{T}$ and the integrals considered are assumed to exist.