New Variants of Quantum Midpoint-Type Inequalities
Recently, there has been a strong push toward creating and expanding quadrature inequalities in quantum calculus. In order to investigate various avenues for quantum inquiry, a number of quantum extensions of midpoint estimations are studied. The goal of this research article is to discover novel qu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/12/2599 |
| _version_ | 1797455196000878592 |
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| author | Saad Ihsan Butt Hüseyin Budak Kamsing Nonlaopon |
| author_facet | Saad Ihsan Butt Hüseyin Budak Kamsing Nonlaopon |
| author_sort | Saad Ihsan Butt |
| collection | DOAJ |
| description | Recently, there has been a strong push toward creating and expanding quadrature inequalities in quantum calculus. In order to investigate various avenues for quantum inquiry, a number of quantum extensions of midpoint estimations are studied. The goal of this research article is to discover novel quantum midpoint-type inequalities that are twice <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="fraktur">q</mi></mrow><msub><mi>ξ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-differentiable for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-convex functions. Firstly, we obtain novel identity for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="fraktur">q</mi></mrow><msub><mi>ξ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-integral by employing quantum calculus tools. Then by using the auxiliary identity, we formulate new bounds by taking into account the known quantum Hölder and Power mean inequalities. An example is provided with a graphical representation to show the validity of obtaining results. The outcomes of this study clarify and expand earlier research on midpoint-type inequalities. Analytic inequalities of this type as well as particularly related strategies have applications for various fields where symmetry plays an important role. |
| first_indexed | 2024-03-09T15:48:02Z |
| format | Article |
| id | doaj.art-c72c3645b7b4483c9dcc412c8f0c8192 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T15:48:02Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-c72c3645b7b4483c9dcc412c8f0c81922023-11-24T18:19:49ZengMDPI AGSymmetry2073-89942022-12-011412259910.3390/sym14122599New Variants of Quantum Midpoint-Type InequalitiesSaad Ihsan Butt0Hüseyin Budak1Kamsing Nonlaopon2Department of Mathematics, Lahore Campus, COMSATS University Islamabad, Islamabad 54000, PakistanDepartment of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, TurkeyDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandRecently, there has been a strong push toward creating and expanding quadrature inequalities in quantum calculus. In order to investigate various avenues for quantum inquiry, a number of quantum extensions of midpoint estimations are studied. The goal of this research article is to discover novel quantum midpoint-type inequalities that are twice <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="fraktur">q</mi></mrow><msub><mi>ξ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-differentiable for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-convex functions. Firstly, we obtain novel identity for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="fraktur">q</mi></mrow><msub><mi>ξ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-integral by employing quantum calculus tools. Then by using the auxiliary identity, we formulate new bounds by taking into account the known quantum Hölder and Power mean inequalities. An example is provided with a graphical representation to show the validity of obtaining results. The outcomes of this study clarify and expand earlier research on midpoint-type inequalities. Analytic inequalities of this type as well as particularly related strategies have applications for various fields where symmetry plays an important role.https://www.mdpi.com/2073-8994/14/12/2599quantum derivativequantum integralquantum midpoint inequalities(<i>α</i>,<i>m</i>)-convex functions |
| spellingShingle | Saad Ihsan Butt Hüseyin Budak Kamsing Nonlaopon New Variants of Quantum Midpoint-Type Inequalities Symmetry quantum derivative quantum integral quantum midpoint inequalities (<i>α</i>,<i>m</i>)-convex functions |
| title | New Variants of Quantum Midpoint-Type Inequalities |
| title_full | New Variants of Quantum Midpoint-Type Inequalities |
| title_fullStr | New Variants of Quantum Midpoint-Type Inequalities |
| title_full_unstemmed | New Variants of Quantum Midpoint-Type Inequalities |
| title_short | New Variants of Quantum Midpoint-Type Inequalities |
| title_sort | new variants of quantum midpoint type inequalities |
| topic | quantum derivative quantum integral quantum midpoint inequalities (<i>α</i>,<i>m</i>)-convex functions |
| url | https://www.mdpi.com/2073-8994/14/12/2599 |
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