On Hermite–Hadamard–Fejér-Type Inequalities for <i>η</i>-Convex Functions via Quantum Calculus
In this paper, we use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mi>a</mi></msub></semantics></math></inline-formula>- and <inline-formula...
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| Format: | Article |
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MDPI AG
2023-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/15/3387 |
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| author | Nuttapong Arunrat Kamsing Nonlaopon Hüseyin Budak |
| author_facet | Nuttapong Arunrat Kamsing Nonlaopon Hüseyin Budak |
| author_sort | Nuttapong Arunrat |
| collection | DOAJ |
| description | In this paper, we use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mi>a</mi></msub></semantics></math></inline-formula>- and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>q</mi><mi>b</mi></msup></semantics></math></inline-formula>-integrals to establish some quantum Hermite–Hadamard–Fejér-type inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions. By taking <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>, our results reduce to classical results on Hermite–Hadamard–Fejér-type inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions. Moreover, we give some examples for quantum Hermite–Hadamard–Fejér-type inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions. Some results presented here for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions provide extensions of others given in earlier works for convex and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions. |
| first_indexed | 2024-03-11T00:22:56Z |
| format | Article |
| id | doaj.art-e5e16c6c27934f4fbd8bb2af7270b4df |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T00:22:56Z |
| publishDate | 2023-08-01 |
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| series | Mathematics |
| spelling | doaj.art-e5e16c6c27934f4fbd8bb2af7270b4df2023-11-18T23:15:58ZengMDPI AGMathematics2227-73902023-08-011115338710.3390/math11153387On Hermite–Hadamard–Fejér-Type Inequalities for <i>η</i>-Convex Functions via Quantum CalculusNuttapong Arunrat0Kamsing Nonlaopon1Hüseyin Budak2Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandDepartment of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, TurkeyIn this paper, we use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mi>a</mi></msub></semantics></math></inline-formula>- and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>q</mi><mi>b</mi></msup></semantics></math></inline-formula>-integrals to establish some quantum Hermite–Hadamard–Fejér-type inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions. By taking <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>, our results reduce to classical results on Hermite–Hadamard–Fejér-type inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions. Moreover, we give some examples for quantum Hermite–Hadamard–Fejér-type inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions. Some results presented here for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions provide extensions of others given in earlier works for convex and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-convex functions.https://www.mdpi.com/2227-7390/11/15/3387Hermite–Hadamard–Fejér-type integral inequalities<i>η</i>-convex functionquantum calculus<i>q<sub>a</sub></i>-integral<i>q<sup>b</sup></i>-integral |
| spellingShingle | Nuttapong Arunrat Kamsing Nonlaopon Hüseyin Budak On Hermite–Hadamard–Fejér-Type Inequalities for <i>η</i>-Convex Functions via Quantum Calculus Mathematics Hermite–Hadamard–Fejér-type integral inequalities <i>η</i>-convex function quantum calculus <i>q<sub>a</sub></i>-integral <i>q<sup>b</sup></i>-integral |
| title | On Hermite–Hadamard–Fejér-Type Inequalities for <i>η</i>-Convex Functions via Quantum Calculus |
| title_full | On Hermite–Hadamard–Fejér-Type Inequalities for <i>η</i>-Convex Functions via Quantum Calculus |
| title_fullStr | On Hermite–Hadamard–Fejér-Type Inequalities for <i>η</i>-Convex Functions via Quantum Calculus |
| title_full_unstemmed | On Hermite–Hadamard–Fejér-Type Inequalities for <i>η</i>-Convex Functions via Quantum Calculus |
| title_short | On Hermite–Hadamard–Fejér-Type Inequalities for <i>η</i>-Convex Functions via Quantum Calculus |
| title_sort | on hermite hadamard fejer type inequalities for i η i convex functions via quantum calculus |
| topic | Hermite–Hadamard–Fejér-type integral inequalities <i>η</i>-convex function quantum calculus <i>q<sub>a</sub></i>-integral <i>q<sup>b</sup></i>-integral |
| url | https://www.mdpi.com/2227-7390/11/15/3387 |
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