Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities
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’ d...
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| Format: | Article |
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MDPI AG
2023-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/1/104 |
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| author | Haytham M. Rezk Juan E. Nápoles Valdés Maha Ali Ahmed I. Saied Mohammed Zakarya |
| author_facet | Haytham M. Rezk Juan E. Nápoles Valdés Maha Ali Ahmed I. Saied Mohammed Zakarya |
| author_sort | Haytham M. Rezk |
| collection | DOAJ |
| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>s</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="4pt"></mspace><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>s</mi><mi>j</mi></msub><mo>,</mo><msub><mi>z</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>w</mi><mi>j</mi></msub><mo>,</mo><msub><mi>n</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>), we obtain the discrete and continuous inequalities. Also, we can obtain other inequalities in different time scales, like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><msup><mi>q</mi><mover><mi mathvariant="double-struck">Z</mi><mo>−</mo></mover></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-08T15:02:37Z |
| format | Article |
| id | doaj.art-9ae92ce66de044fe97759c8aeca87cf7 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T15:02:37Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9ae92ce66de044fe97759c8aeca87cf72024-01-10T15:03:37ZengMDPI AGMathematics2227-73902023-12-0112110410.3390/math12010104Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type InequalitiesHaytham M. Rezk0Juan E. Nápoles Valdés1Maha Ali2Ahmed I. Saied3Mohammed Zakarya4Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, EgyptFacultad de Ciencias Exactas y Naturales y Agrimensura, Universidad Nacional del Nordeste, Av. Libertad 5450, Corrientes 3400, ArgentinaDepartment of Mathematics, College of Arts and Sciences, King Khalid University, P.O. Box 64512, Abha 62529, Sarat Ubaidah, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Benha University, Benha 13511, EgyptDepartment of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi ArabiaIn 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>s</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mspace width="4pt"></mspace><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>s</mi><mi>j</mi></msub><mo>,</mo><msub><mi>z</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>a</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>w</mi><mi>j</mi></msub><mo>,</mo><msub><mi>n</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∀</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>), we obtain the discrete and continuous inequalities. Also, we can obtain other inequalities in different time scales, like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">T</mi><mo>=</mo><msup><mi>q</mi><mover><mi mathvariant="double-struck">Z</mi><mo>−</mo></mover></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/12/1/104Hilbert-type inequalitiesHölder’s inequalitymean inequalitykernelsdelta integralstime scales |
| spellingShingle | Haytham M. Rezk Juan E. Nápoles Valdés Maha Ali Ahmed I. Saied Mohammed Zakarya Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities Mathematics Hilbert-type inequalities Hölder’s inequality mean inequality kernels delta integrals time scales |
| title | Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities |
| title_full | Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities |
| title_fullStr | Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities |
| title_full_unstemmed | Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities |
| title_short | Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities |
| title_sort | delta calculus on time scale formulas that are similar to hilbert type inequalities |
| topic | Hilbert-type inequalities Hölder’s inequality mean inequality kernels delta integrals time scales |
| url | https://www.mdpi.com/2227-7390/12/1/104 |
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