Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in <i>q</i>-Calculus
The main objective of this study is to establish two important right <i>q</i>-integral equalities involving a right-quantum derivative with parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mro...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-01-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/10/3/444 |
| _version_ | 1797486385706303488 |
|---|---|
| author | Dafang Zhao Ghazala Gulshan Muhammad Aamir Ali Kamsing Nonlaopon |
| author_facet | Dafang Zhao Ghazala Gulshan Muhammad Aamir Ali Kamsing Nonlaopon |
| author_sort | Dafang Zhao |
| collection | DOAJ |
| description | The main objective of this study is to establish two important right <i>q</i>-integral equalities involving a right-quantum derivative with parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-convex functions. The fundamental benefit of these inequalities is that they may be transformed into <i>q</i>-midpoint- and <i>q</i>-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field. |
| first_indexed | 2024-03-09T23:32:25Z |
| format | Article |
| id | doaj.art-b4c1dd430f494f26909923c40519cd1c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T23:32:25Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b4c1dd430f494f26909923c40519cd1c2023-11-23T17:07:36ZengMDPI AGMathematics2227-73902022-01-0110344410.3390/math10030444Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in <i>q</i>-CalculusDafang Zhao0Ghazala Gulshan1Muhammad Aamir Ali2Kamsing Nonlaopon3School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, ChinaDepartment of Mathematics, Faculty of Science, Mirpur University of Science and Technology (MUST), Mirpur 10250, PakistanJiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, ChinaDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandThe main objective of this study is to establish two important right <i>q</i>-integral equalities involving a right-quantum derivative with parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. Then, utilizing these equalities, we derive some new variants for midpoint- and trapezoid-type inequalities for the right-quantum integral via differentiable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-convex functions. The fundamental benefit of these inequalities is that they may be transformed into <i>q</i>-midpoint- and <i>q</i>-trapezoid-type inequalities for convex functions, classical midpoint inequalities for convex functions and classical trapezoid-type inequalities for convex functions are transformed without having to prove each one independently. In addition, we present some applications of our results to special means of positive real numbers. It is expected that the ideas and techniques may stimulate further research in this field.https://www.mdpi.com/2227-7390/10/3/444midpoint inequalitiestrapezoid inequalitiesquantum calculus(<i>α</i>, <i>m</i>)-convex functions |
| spellingShingle | Dafang Zhao Ghazala Gulshan Muhammad Aamir Ali Kamsing Nonlaopon Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in <i>q</i>-Calculus Mathematics midpoint inequalities trapezoid inequalities quantum calculus (<i>α</i>, <i>m</i>)-convex functions |
| title | Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in <i>q</i>-Calculus |
| title_full | Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in <i>q</i>-Calculus |
| title_fullStr | Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in <i>q</i>-Calculus |
| title_full_unstemmed | Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in <i>q</i>-Calculus |
| title_short | Some New Midpoint and Trapezoidal-Type Inequalities for General Convex Functions in <i>q</i>-Calculus |
| title_sort | some new midpoint and trapezoidal type inequalities for general convex functions in i q i calculus |
| topic | midpoint inequalities trapezoid inequalities quantum calculus (<i>α</i>, <i>m</i>)-convex functions |
| url | https://www.mdpi.com/2227-7390/10/3/444 |
| work_keys_str_mv | AT dafangzhao somenewmidpointandtrapezoidaltypeinequalitiesforgeneralconvexfunctionsiniqicalculus AT ghazalagulshan somenewmidpointandtrapezoidaltypeinequalitiesforgeneralconvexfunctionsiniqicalculus AT muhammadaamirali somenewmidpointandtrapezoidaltypeinequalitiesforgeneralconvexfunctionsiniqicalculus AT kamsingnonlaopon somenewmidpointandtrapezoidaltypeinequalitiesforgeneralconvexfunctionsiniqicalculus |