On Some New Dynamic Hilbert-Type Inequalities Across Time Scales

Zakarya, Mohammed; Saied, Ahmed I.; Al-Thaqfan, Amirah Ayidh I; Ali, Maha; Rezk, Haytham M.

doi:10.3390/axioms13070475

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On Some New Dynamic Hilbert-Type Inequalities Across Time Scales

by

Mohammed Zakarya

¹

,

Ahmed I. Saied

^2,3,

Amirah Ayidh I Al-Thaqfan

⁴,

Maha Ali

⁴ and

Haytham M. Rezk

^5,*

¹

Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia

²

Mathematical Institute, Slovak Academy of Sciences, Grekošákova 6, 04001 Košice, Slovakia

³

Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt

⁴

Department of Mathematics, College of Arts and Sciences, King Khalid University, P.O. Box 64512, Abha 62529, Sarat Ubaidah, Saudi Arabia

⁵

Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt

^*

Author to whom correspondence should be addressed.

Axioms 2024, 13(7), 475; https://doi.org/10.3390/axioms13070475

Submission received: 19 May 2024 / Revised: 29 June 2024 / Accepted: 12 July 2024 / Published: 14 July 2024

(This article belongs to the Special Issue Infinite Dynamical System and Differential Equations)

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Abstract

In this article, we present some novel dynamic Hilbert-type inequalities within the framework of time scales

T

. We achieve this by utilizing Hölder’s inequality, the chain rule, and the mean inequality. As specific instances of our findings (when

T = N

and

T = R

), we obtain the discrete and continuous analogues of previously established inequalities. Additionally, we derive other inequalities for different time scales, such as

T = q^{N_{0}}

for

q > 1,

which, to the best of the authors’ knowledge, is a largely novel conclusion.

Keywords: Hilbert-type inequalities; Hölder’s inequality; chain rule; mean inequality; dynamic inequality; time scale delta calculus

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MDPI and ACS Style

Zakarya, M.; Saied, A.I.; Al-Thaqfan, A.A.I.; Ali, M.; Rezk, H.M. On Some New Dynamic Hilbert-Type Inequalities Across Time Scales. Axioms 2024, 13, 475. https://doi.org/10.3390/axioms13070475

AMA Style

Zakarya M, Saied AI, Al-Thaqfan AAI, Ali M, Rezk HM. On Some New Dynamic Hilbert-Type Inequalities Across Time Scales. Axioms. 2024; 13(7):475. https://doi.org/10.3390/axioms13070475

Chicago/Turabian Style

Zakarya, Mohammed, Ahmed I. Saied, Amirah Ayidh I Al-Thaqfan, Maha Ali, and Haytham M. Rezk. 2024. "On Some New Dynamic Hilbert-Type Inequalities Across Time Scales" Axioms 13, no. 7: 475. https://doi.org/10.3390/axioms13070475

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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On Some New Dynamic Hilbert-Type Inequalities Across Time Scales

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