New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with <i>ψ</i><sub><i>k</i></sub>-Raina’s Fractional Integrals for Differentiable Convex Functions
Starting from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Raina’s fractional integrals (&...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-03-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/9/4/203 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850180418678030336 |
|---|---|
| author | Talib Hussain Loredana Ciurdariu Eugenia Grecu |
| author_facet | Talib Hussain Loredana Ciurdariu Eugenia Grecu |
| author_sort | Talib Hussain |
| collection | DOAJ |
| description | Starting from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Raina’s fractional integrals (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), <i>k</i>-Riemann–Liouville (<i>k</i>-R-L), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Riemann–Liouville (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-R-L), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Riemann–Liouville (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-R-L), Raina’s, <i>k</i>-Raina’s, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Raina’s fractional integrals (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function. |
| format | Article |
| id | doaj-art-dacf1e44c8df4d0ab3eca1c4ff4969b7 |
| institution | OA Journals |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-dacf1e44c8df4d0ab3eca1c4ff4969b72025-08-20T02:18:11ZengMDPI AGFractal and Fractional2504-31102025-03-019420310.3390/fractalfract9040203New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with <i>ψ</i><sub><i>k</i></sub>-Raina’s Fractional Integrals for Differentiable Convex FunctionsTalib Hussain0Loredana Ciurdariu1Eugenia Grecu2Department of Mathematics and Statistics, University of Agriculture Faisalabad, Faisalabad 38000, PakistanDepartment of Mathematics, Politehnica University of Timișoara, 300006 Timisoara, RomaniaDepartment of Management, Politehnica University of Timișoara, 300006 Timisoara, RomaniaStarting from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Raina’s fractional integrals (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-RFIs), the study obtains a new generalization of the Hermite–Hadamard–Mercer (H-H-M) inequality. Several trapezoid-type inequalities are constructed for functions whose derivatives of orders 1 and 2, in absolute value, are convex and involve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-RFIs. The results of the research are refinements of the Hermite–Hadamard (H-H) and H-H-M-type inequalities. For several types of fractional integrals—Riemann–Liouville (R-L), <i>k</i>-Riemann–Liouville (<i>k</i>-R-L), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Riemann–Liouville (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-R-L), <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Riemann–Liouville (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ψ</mi><mi>k</mi></msub></semantics></math></inline-formula>-R-L), Raina’s, <i>k</i>-Raina’s, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Raina’s fractional integrals (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-RFIs)—new inequalities of H-H and H-H-M-type are established, respectively. This article presents special cases of the main results and provides numerous examples with graphical illustrations to confirm the validity of the results. This study shows the efficiency of the findings with a couple of applications, taking into account the modified Bessel function and the q-digamma function.https://www.mdpi.com/2504-3110/9/4/203<i>ψ<sub>k</sub></i>-Raina’s fractional integralsHermite–Hadamard–Mercer inequalityJensen–Mercer inequalityconvex functionBessel function<i>q</i>-digamma function |
| spellingShingle | Talib Hussain Loredana Ciurdariu Eugenia Grecu New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with <i>ψ</i><sub><i>k</i></sub>-Raina’s Fractional Integrals for Differentiable Convex Functions Fractal and Fractional <i>ψ<sub>k</sub></i>-Raina’s fractional integrals Hermite–Hadamard–Mercer inequality Jensen–Mercer inequality convex function Bessel function <i>q</i>-digamma function |
| title | New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with <i>ψ</i><sub><i>k</i></sub>-Raina’s Fractional Integrals for Differentiable Convex Functions |
| title_full | New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with <i>ψ</i><sub><i>k</i></sub>-Raina’s Fractional Integrals for Differentiable Convex Functions |
| title_fullStr | New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with <i>ψ</i><sub><i>k</i></sub>-Raina’s Fractional Integrals for Differentiable Convex Functions |
| title_full_unstemmed | New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with <i>ψ</i><sub><i>k</i></sub>-Raina’s Fractional Integrals for Differentiable Convex Functions |
| title_short | New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with <i>ψ</i><sub><i>k</i></sub>-Raina’s Fractional Integrals for Differentiable Convex Functions |
| title_sort | new perspectives of hermite hadamard mercer type inequalities associated with i ψ i sub i k i sub raina s fractional integrals for differentiable convex functions |
| topic | <i>ψ<sub>k</sub></i>-Raina’s fractional integrals Hermite–Hadamard–Mercer inequality Jensen–Mercer inequality convex function Bessel function <i>q</i>-digamma function |
| url | https://www.mdpi.com/2504-3110/9/4/203 |
| work_keys_str_mv | AT talibhussain newperspectivesofhermitehadamardmercertypeinequalitiesassociatedwithipsisubikisubrainasfractionalintegralsfordifferentiableconvexfunctions AT loredanaciurdariu newperspectivesofhermitehadamardmercertypeinequalitiesassociatedwithipsisubikisubrainasfractionalintegralsfordifferentiableconvexfunctions AT eugeniagrecu newperspectivesofhermitehadamardmercertypeinequalitiesassociatedwithipsisubikisubrainasfractionalintegralsfordifferentiableconvexfunctions |