Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus
In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semanti...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-07-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/9/14/1666 |
| _version_ | 1797526686545215488 |
|---|---|
| author | Surang Sitho Muhammad Aamir Ali Hüseyin Budak Sotiris K. Ntouyas Jessada Tariboon |
| author_facet | Surang Sitho Muhammad Aamir Ali Hüseyin Budak Sotiris K. Ntouyas Jessada Tariboon |
| author_sort | Surang Sitho |
| collection | DOAJ |
| description | In this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>q</mi><msub><mi>ϰ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-quantum integral to show midpoint and trapezoidal inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>q</mi><msub><mi>ϰ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-quantum integral is not valid for preinvex functions, and we present its proper form. We use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-quantum integrals to show midpoint inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-differentiable preinvex functions. It is also demonstrated that by considering the limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>→</mo><msup><mn>1</mn><mo>−</mo></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mfenced separators="" open="(" close=")"><msub><mi>ϰ</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ϰ</mi><mn>1</mn></msub></mfenced><mo>=</mo><mo>−</mo><mi>η</mi><mfenced separators="" open="(" close=")"><msub><mi>ϰ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ϰ</mi><mn>2</mn></msub></mfenced><mo>=</mo><msub><mi>ϰ</mi><mn>2</mn></msub><mo>−</mo><msub><mi>ϰ</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> in the newly derived results, the newly proved findings can be turned into certain known results. |
| first_indexed | 2024-03-10T09:32:52Z |
| format | Article |
| id | doaj.art-e7ef8baf3e454ef39a18609d51325ac0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T09:32:52Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-e7ef8baf3e454ef39a18609d51325ac02023-11-22T04:20:18ZengMDPI AGMathematics2227-73902021-07-01914166610.3390/math9141666Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum CalculusSurang Sitho0Muhammad Aamir Ali1Hüseyin Budak2Sotiris K. Ntouyas3Jessada Tariboon4Department of Social and Applied Science, College of Industrial Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, ThailandJiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, ChinaDepartment of Mathematics, Faculty of Science and Arts, Düzce University, Düzce 81620, TurkeyDepartment of Mathematics, University of Ioannina, 45110 Ioannina, GreeceIntelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, ThailandIn this article, we use quantum integrals to derive Hermite–Hadamard inequalities for preinvex functions and demonstrate their validity with mathematical examples. We use the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>q</mi><msub><mi>ϰ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-quantum integral to show midpoint and trapezoidal inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>q</mi><msub><mi>ϰ</mi><mn>2</mn></msub></msup></semantics></math></inline-formula>-differentiable preinvex functions. Furthermore, we demonstrate with an example that the previously proved Hermite–Hadamard-type inequality for preinvex functions via <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-quantum integral is not valid for preinvex functions, and we present its proper form. We use <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-quantum integrals to show midpoint inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><msub><mi>ϰ</mi><mn>1</mn></msub></msub></semantics></math></inline-formula>-differentiable preinvex functions. It is also demonstrated that by considering the limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>→</mo><msup><mn>1</mn><mo>−</mo></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mfenced separators="" open="(" close=")"><msub><mi>ϰ</mi><mn>2</mn></msub><mo>,</mo><msub><mi>ϰ</mi><mn>1</mn></msub></mfenced><mo>=</mo><mo>−</mo><mi>η</mi><mfenced separators="" open="(" close=")"><msub><mi>ϰ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ϰ</mi><mn>2</mn></msub></mfenced><mo>=</mo><msub><mi>ϰ</mi><mn>2</mn></msub><mo>−</mo><msub><mi>ϰ</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula> in the newly derived results, the newly proved findings can be turned into certain known results.https://www.mdpi.com/2227-7390/9/14/1666Hermite–Hadamard inequalityq-integralquantum calculuspreinvex functiontrapezoid inequalitiesmidpoint inequalities |
| spellingShingle | Surang Sitho Muhammad Aamir Ali Hüseyin Budak Sotiris K. Ntouyas Jessada Tariboon Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus Mathematics Hermite–Hadamard inequality q-integral quantum calculus preinvex function trapezoid inequalities midpoint inequalities |
| title | Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus |
| title_full | Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus |
| title_fullStr | Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus |
| title_full_unstemmed | Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus |
| title_short | Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus |
| title_sort | trapezoid and midpoint type inequalities for preinvex functions via quantum calculus |
| topic | Hermite–Hadamard inequality q-integral quantum calculus preinvex function trapezoid inequalities midpoint inequalities |
| url | https://www.mdpi.com/2227-7390/9/14/1666 |
| work_keys_str_mv | AT surangsitho trapezoidandmidpointtypeinequalitiesforpreinvexfunctionsviaquantumcalculus AT muhammadaamirali trapezoidandmidpointtypeinequalitiesforpreinvexfunctionsviaquantumcalculus AT huseyinbudak trapezoidandmidpointtypeinequalitiesforpreinvexfunctionsviaquantumcalculus AT sotiriskntouyas trapezoidandmidpointtypeinequalitiesforpreinvexfunctionsviaquantumcalculus AT jessadatariboon trapezoidandmidpointtypeinequalitiesforpreinvexfunctionsviaquantumcalculus |