Some Dynamic Inequalities of Hilbert’s Type

A. M. Ahmed , Ghada AlNemer, M. Zakarya , and H. M. Rezk Mathematics Department, College of Science, Jouf University, Sakaka (2014), Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 105862, Riyadh 11656, Saudi Arabia King Khalid University, College of Science, Department of Mathematics, P.O. Box 9004, Z 61413 Abha, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azhar University, Z 71524 Assiut, Egypt


Introduction
It is evident that the Hilbert-type inequalities outplay a major role in mathematics, for pattern complex analysis, numerical analysis, and qualitative theory of differential equations and their implementations. In recent years, there were a lot of various refinements, generalizations, extensions, and applications of Hilbert's inequality which have seemed in the literature. Hilbert's discrete inequality and its integral formula ( [1], eorem 316) have been generalized in many trends (for example, see [2][3][4][5][6]). Lately, Pachpatte [7] proved new inequalities similar to those of Hilbert's inequality, namely, he proved that if h, l ≥ 1, A m � m s�1 a s ≥ 0, and B n � n t�1 b t ≥ 0, then where C * (h, l, k, r) � 1 2 hl �� kr An integral analogue of (1) is given in the following result. Let h, l ≥ 1, F(x) � x 0 f(τ)dτ ≥ 0, and G(y) � y 0 g(])d] ≥ 0, for x, τ ∈ (0, a) and y, ] ∈ (0, b). en, where where h, l ≥ 1, A m � m p�1 a p ≥ 0, B n � n q�1 b q ≥ 0, and C(h, l, k, r, α) � 1 2 An integral analogue of (5) is given in the following result. Let p, q ≥ 1, α > 0, where In 2009, Yang [9] gave another generalization of (1) and (3) by introducing parameter α > 1 and c > 1 as follows. Let h, l ≥ 1, A m � m p�1 a p ≥ 0, and B n � n q�1 b q ≥ 0. en, where An integral analogue of (9) is given as follows. If h, where In [10], the authors deduced several generalizations of inequalities (1) and (3) on time scales, namely, they proved that if h and l ≥ 1 are real numbers, where In [11], the authors gave some extensions of inequalities (5) and (7) on time scales. Minutely, they proved that if c > 0 and h and l ≥ 1 are real numbers, where C(h, l, η, c) � hl 1 2 2/ηc Following this trend and to develop the study of dynamic inequalities on time scales, we will prove some new inequalities of Hilbert's type on time scales, namely, we prove time scale versions of inequalities (9) and (11) on time scale T. ese inequalities can be considered as extensions and generalizations of some Hilbert-type inequalities proved in [10]. We also derive some inequalities on time scale as special cases.

Definitions and Basic Results
In this division, we will present some fundamental concepts and effects on time scales which will be beneficial for deducing our main results. e following definitions and theorems are referred from [12,13].
Time scale T is defined as a nonempty arbitrary locked subplot of real numbers R. We define the forward jump operator σ: T ⟶ T as and the backward jump operator ρ: T ⟶ T as From the above two definitions, it can be stated that a For a function χ: T ⟶ R, the delta derivative of χ at τ ∈ T k is defined as for each ε > 0, there is a neighborhood U of τ such that A function χ: T ⟶ R is called right-dense continuous (rd−continuous) as long as it is continuous at all right-dense points in T, and its left-sided limits exist (finite) at all leftdense points in T. e classes of real rd−continuous functions on an interval I will be denoted by C rd (T, R). For θ, t ∈ T, the Cauchy integral of χ Δ is defined as Note that In what follows, we will present Hölder's inequality, Jensen's inequality, and the power rules for integration on time scales.
Theorem 2 (Jensen's inequality (see [14,16] Lemma 3 (see [17]). Let u, s ∈ T, and ζ Now, we will present the formula that will reduce double integrals to single integrals which is the desired in [18].
assuming the integrals considered exist.

Corollary 8. Assume that {a i } and {b j } are two nonnegative sequences of real numbers, and define
where which was proved by Yang ([9], eorem 2.1).
en, for s ∈ (0, x) and θ ∈ (0, y), we have where It is clear that we can have the same inequality in [9], eorem 3.2. By using relations (23) and putting T � Z and t 0 � 0 in eorem 9, we get the following conclusion.

Corollary 14.
Assume that {a i } and {b j } are two nonnegative sequences of real numbers, and define