Citation: Zakarya, M.; AlNemer, G.; Saied, A.I.; Butush, R.; Bazighifan, O.; Rezk, H.M. Generalized Inequalities of Hilbert-Type on Time Scales Nabla Calculus. Symmetry 2022, 14, 1512. https://doi.org/10.3390/ sym14081512 Academic Editor: Ioan Ras , a Received: 27 May 2022 Accepted: 5 July 2022 Published: 24 July 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). symmetry SS Article Generalized Inequalities of Hilbert-Type on Time Scales Nabla Calculus Mohammed Zakarya 1,2, * , Ghada AlNemer 3, * , Ahmed I. Saied 4 , Roqia Butush 5 , Omar Bazighifan 6 and Haytham M. Rezk 7 1 Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt 3 Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt; as0863289@gmail.com 5 Department of Mathematics, University of Jordan, Amman P.O. Box 11941, Jordan; butushroqia@gmail.com 6 Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy; o.bazighifan@gmail.com 7 Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt; haythamrezk@azhar.edu.eg * Correspondence: mzibrahim@kku.edu.sa (M.Z.); gnnemer@pnu.edu.sa (G.A.) Abstract: In this paper, we prove some new generalized inequalities of Hilbert-type on time scales nabla calculus by applying Hölder’s inequality, Young’s inequality, and Jensen’s inequality. Symmet- rical properties play an essential role in determining the correct methods to solve inequalities. Keywords: Hilbert-type inequalities; time scales nabla calculus; Hölder’s inequality; Young’s inequality 1. Introduction 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 {β m } and {d n } are two real sequences such that 0 < ∑ ∞ m=1 β 2 m < ∞ and 0 < ∑ ∞ n=1 d 2 n < ∞, then ∞ ∑ n=1 ∞ ∑ m=1 β m d n m + n ≤ π  ∞ ∑ m=1 β 2 m ! 1 2  ∞ ∑ n=1 d 2 n ! 1 2 . (1) In 1911, Schur [1] proved that π in (1) is sharp and also discovered the integral analogue of (1), which became known as the Hilbert integral inequality in the form Z ∞ 0 Z ∞ 0 Ξ(τ)Υ(y) τ + y dτdy ≤ π  Z ∞ 0 Ξ 2 (τ)dτ  1 2  Z ∞ 0 Υ 2 (y)dy  1 2 , (2) where Ξ and Υ are measurable functions such that, 0 < R ∞ 0 Ξ 2 (τ)dτ < ∞ and 0 < R ∞ 0 Υ 2 (y)dy < ∞. 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, β m , d n ≥ 0 such that 0 < ∑ ∞ m=1 β p m < ∞ and 0 < ∑ ∞ n=1 d q n < ∞, then ∞ ∑ n=1 ∞ ∑ m=1 β m d n m + n ≤ π sin π p  ∞ ∑ m=1 β p m ! 1 p  ∞ ∑ n=1 d q n ! 1 q , (3) Symmetry 2022, 14, 1512. https://doi.org/10.3390/sym14081512 https://www.mdpi.com/journal/symmetry