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Open AccessArticle

New Perspectives of Hermite–Hadamard–Mercer-Type Inequalities Associated with ψ_k-Raina’s Fractional Integrals for Differentiable Convex Functions

by Talib Hussain, Loredana Ciurdariu and Eugenia Grecu

Fractal Fract. 2025, 9(4), 203; https://doi.org/10.3390/fractalfract9040203 - 26 Mar 2025

Cited by 2 | Viewed by 756

Abstract

Starting from

ψ_{k}

-Raina’s fractional integrals (

ψ_{k}

-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 [...] Read more.

Starting from

ψ_{k}

-Raina’s fractional integrals (

ψ_{k}

ψ_{k}

-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), k-Riemann–Liouville (k-R-L),

ψ

-Riemann–Liouville (

ψ

-R-L),

ψ_{k}

-Riemann–Liouville (

ψ_{k}

-R-L), Raina’s, k-Raina’s, and

ψ

-Raina’s fractional integrals (

ψ

-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. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)

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12 pages, 344 KB

Open AccessArticle

New Approximation Formula of Digamma Function with Bounded Remainder

by Mansour Mahmoud, Abdulaziz S. Alofi and Mohammed A. Zurayyir

Mathematics 2025, 13(5), 720; https://doi.org/10.3390/math13050720 - 24 Feb 2025

Viewed by 2507

Abstract

This study establishes the new approximation formula for the Digamma function [...] Read more.

This study establishes the new approximation formula for the Digamma function

ψ (s) = ln s - \frac{1}{2 s} - \frac{1}{12 s \sqrt{s^{2} + θ (s)}}, \frac{1}{36} < θ (s) < \frac{1}{5}; s > 0

, as well as some of its inequalities, where

θ (s)

is a continuous function. We demonstrate numerically that our results are superior to some recent results. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis, Analytic Number Theory, and Mathematical Inequalities)

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26 pages, 401 KB

Open AccessArticle

Results on Minkowski-Type Inequalities for Weighted Fractional Integral Operators

by Hari Mohan Srivastava, Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Artion Kashuri and Nejmeddine Chorfi

Symmetry 2023, 15(8), 1522; https://doi.org/10.3390/sym15081522 - 2 Aug 2023

Cited by 9 | Viewed by 2397

Abstract

This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful methods to help with the learning of key mathematical ideas. The kernel of the general family of weighted fractional integral operators is related to a wide variety of extensions and generalizations of the Mittag-Leffler function and the Hurwitz-Lerch zeta function. It delves into the applications of fractional-order integral and derivative operators in mathematical and engineering sciences. Furthermore, this article derives specific cases for selected functions and presents various applications to illustrate the obtained results. Additionally, novel applications involving the Digamma function are introduced. Full article

(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)

13 pages, 307 KB

Open AccessArticle

An Asymptotic Expansion for the Generalized Gamma Function

by Mansour Mahmoud, Hanan Almuashi and Hesham Moustafa

Symmetry 2022, 14(7), 1412; https://doi.org/10.3390/sym14071412 - 9 Jul 2022

Cited by 4 | Viewed by 2179

Abstract

The symmetric patterns that inequalities contain are reflected in researchers’ studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function

Γ_{μ} (v)

and study the completely monotonic (CM) property of a function [...] Read more.

Γ_{μ} (v)

and study the completely monotonic (CM) property of a function involving

Γ_{μ} (v)

and the generalized digamma function

ψ_{μ} (v)

. As a consequence, we establish some bounds for

Γ_{μ} (v)

ψ_{μ} (v)

and polygamma functions

ψ_{μ}^{(r)} (v)

r \geq 1

. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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