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

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 (&...

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Bibliographic Details
Main Authors:	Talib Hussain, Loredana Ciurdariu, Eugenia Grecu
Format:	Article
Language:	English
Published:	MDPI AG 2025-03-01
Series:	Fractal and Fractional
Subjects:	<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
Online Access:	https://www.mdpi.com/2504-3110/9/4/203
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Summary:	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.
ISSN:	2504-3110