A. I. Saied - Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovakia. - Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt. I. Jadlovská - Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, 040 01 Košice, Slovakia.
In this paper, we establish new generalizations of certain Hilbert-type dynamic inequalities on an arbitrary time scale \(\mathbb{T}\), by constructing a general kernel in \(n\)-dimensions for Hilbert-type inequalities. The proofs of our results are based on applications of Hölder's inequality, Fubini's theorem, the integration by parts formula in the nabla calculus on time scales, and the mean inequality. As special cases of our results (corresponding to \(\mathbb{T} = \mathbb{N}\) and \(\mathbb{T} = \mathbb{R}\)), one recovers the classical discrete and continuous inequalities previously established by Yang \cite{yang2008some}. In the setting of quantum calculus (when \(\mathbb{T} = q^{\mathbb{N}_{0}}\) with \(q > 1\)), the resulting inequalities are essentially new.
A. I. Saied, I. Jadlovská, A study on generalized dynamic inequalities of Hilbert-type on time scales, Journal of Mathematics and Computer Science, 41 (2026), no. 4, 535--549
Saied A. I., Jadlovská I., A study on generalized dynamic inequalities of Hilbert-type on time scales. J Math Comput SCI-JM. (2026); 41(4):535--549
Saied, A. I., Jadlovská, I.. "A study on generalized dynamic inequalities of Hilbert-type on time scales." Journal of Mathematics and Computer Science, 41, no. 4 (2026): 535--549
