Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities

Rezk, Haytham M.; Valdés, Juan E. Nápoles; Ali, Maha; Saied, Ahmed I.; Zakarya, Mohammed

doi:10.3390/math12010104

Open AccessArticle

Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities

by

Haytham M. Rezk

¹

,

Juan E. Nápoles Valdés

^2,*

,

Maha Ali

³,

Ahmed I. Saied

⁴ and

Mohammed Zakarya

⁵

¹

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

²

Facultad de Ciencias Exactas y Naturales y Agrimensura, Universidad Nacional del Nordeste, Av. Libertad 5450, Corrientes 3400, Argentina

³

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

⁴

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

⁵

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

^*

Author to whom correspondence should be addressed.

Mathematics 2024, 12(1), 104; https://doi.org/10.3390/math12010104

Submission received: 15 October 2023 / Revised: 24 November 2023 / Accepted: 4 December 2023 / Published: 28 December 2023

(This article belongs to the Section E: Applied Mathematics)

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Abstract

In this article, we establish some new generalized inequalities of the Hilbert-type on time scales’ delta calculus, which can be considered similar to formulas for inequalities of Hilbert type. The major innovation point is to establish some dynamic inequalities of the Hilbert-type on time scales’ delta calculus for delta differentiable functions of one variable and two variables. In this paper, we use the condition

a_{j} (s_{j}) = 0

and

a_{j} (s_{j}, z_{j}) = a_{j} (w_{j}, n_{j}) = 0

,

\forall j = 1, 2, \dots, n

. These inequalities will be proved by applying Hölder’s inequality, the chain rule on time scales, and the mean inequality. As special cases of our results (when

T = N

and

T = R

), we obtain the discrete and continuous inequalities. Also, we can obtain other inequalities in different time scales, like

T = q^{\bar{Z}}

,

q > 1

.

Keywords: Hilbert-type inequalities; Hölder’s inequality; mean inequality; kernels; delta integrals; time scales

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MDPI and ACS Style

Rezk, H.M.; Valdés, J.E.N.; Ali, M.; Saied, A.I.; Zakarya, M. Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities. Mathematics 2024, 12, 104. https://doi.org/10.3390/math12010104

AMA Style

Rezk HM, Valdés JEN, Ali M, Saied AI, Zakarya M. Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities. Mathematics. 2024; 12(1):104. https://doi.org/10.3390/math12010104

Chicago/Turabian Style

Rezk, Haytham M., Juan E. Nápoles Valdés, Maha Ali, Ahmed I. Saied, and Mohammed Zakarya. 2024. "Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities" Mathematics 12, no. 1: 104. https://doi.org/10.3390/math12010104

APA Style

Rezk, H. M., Valdés, J. E. N., Ali, M., Saied, A. I., & Zakarya, M. (2024). Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities. Mathematics, 12(1), 104. https://doi.org/10.3390/math12010104

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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Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities

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