Novel Integral Inequalities on Nabla Time Scales with <i>C</i>-Monotonic Functions
Through the paper, we present several inequalities involving <i>C</i>-monotonic functions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>≥</mo><mn...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/6/1248 |
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| author | Mohammed Zakarya A. I. Saied Maha Ali Haytham M. Rezk Mohammed R. Kenawy |
| author_facet | Mohammed Zakarya A. I. Saied Maha Ali Haytham M. Rezk Mohammed R. Kenawy |
| author_sort | Mohammed Zakarya |
| collection | DOAJ |
| description | Through the paper, we present several inequalities involving <i>C</i>-monotonic functions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will be proved by applying the chain rule formula on nabla calculus. As a special case for 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">R</mi><mo>,</mo></mrow></semantics></math></inline-formula> we obtain the continuous analouges of inequalities that had previously been proved in the literature. 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>, the results, to the best of the authors’ knowledge, are essentially new. Symmetrical properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow></semantics></math></inline-formula>-monotonic functions are critical in determining the best way to solve inequalities. |
| first_indexed | 2024-03-11T01:52:55Z |
| format | Article |
| id | doaj.art-3f5ca0eb12c54b7b94a79366bc3a7e60 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T01:52:55Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-3f5ca0eb12c54b7b94a79366bc3a7e602023-11-18T12:51:31ZengMDPI AGSymmetry2073-89942023-06-01156124810.3390/sym15061248Novel Integral Inequalities on Nabla Time Scales with <i>C</i>-Monotonic FunctionsMohammed Zakarya0A. I. Saied1Maha Ali2Haytham M. Rezk3Mohammed R. Kenawy4Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Benha University, Benha 13511, EgyptDepartment of Mathematics, College of Arts and Sciences, Sarat Abidah, King Khalid University, P.O. Box 64512, Abha 62529, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, EgyptDepartment of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, EgyptThrough the paper, we present several inequalities involving <i>C</i>-monotonic functions with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will be proved by applying the chain rule formula on nabla calculus. As a special case for 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">R</mi><mo>,</mo></mrow></semantics></math></inline-formula> we obtain the continuous analouges of inequalities that had previously been proved in the literature. 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>, the results, to the best of the authors’ knowledge, are essentially new. Symmetrical properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi></mrow></semantics></math></inline-formula>-monotonic functions are critical in determining the best way to solve inequalities.https://www.mdpi.com/2073-8994/15/6/1248<i>C</i>-monotonic functionstime scalesnabla calculuschain rule on nabla calculusinequalities |
| spellingShingle | Mohammed Zakarya A. I. Saied Maha Ali Haytham M. Rezk Mohammed R. Kenawy Novel Integral Inequalities on Nabla Time Scales with <i>C</i>-Monotonic Functions Symmetry <i>C</i>-monotonic functions time scales nabla calculus chain rule on nabla calculus inequalities |
| title | Novel Integral Inequalities on Nabla Time Scales with <i>C</i>-Monotonic Functions |
| title_full | Novel Integral Inequalities on Nabla Time Scales with <i>C</i>-Monotonic Functions |
| title_fullStr | Novel Integral Inequalities on Nabla Time Scales with <i>C</i>-Monotonic Functions |
| title_full_unstemmed | Novel Integral Inequalities on Nabla Time Scales with <i>C</i>-Monotonic Functions |
| title_short | Novel Integral Inequalities on Nabla Time Scales with <i>C</i>-Monotonic Functions |
| title_sort | novel integral inequalities on nabla time scales with i c i monotonic functions |
| topic | <i>C</i>-monotonic functions time scales nabla calculus chain rule on nabla calculus inequalities |
| url | https://www.mdpi.com/2073-8994/15/6/1248 |
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