Muhammad Tariq

57214897969

Publications - 8

Some New Notions of Mathematical Integral Inequalities: Theory and Applications

Maggie Aphane Muhammad Tariq Hijaz Ahmad Ali Asghar Waqar Afzal Rakib Efendiev

Publication Name: International Journal of Analysis and Applications

Publication Date: 2026-01-01

Volume: 24

Issue: Unknown

Page Range: Unknown

Description:

Convex analysis and mathematical inequalities play a fundamental role in both pure and applied sciences. In this work, we first explore the notion of n-fractional polynomial s-like m-convexity involving Raina’s mapping and also its algebraic properties. We then introduce a novel Hermite–Hadamard (H-H), midpoint H-H, trapezoid H-H type inequalities based on this generalized concept and the k-fractional operator. Several related corollaries and examples are examined, particularly in connection with the Mittag–Leffler function. The practical utility of the proposed inequalities is demonstrated through applications to viscoelastic materials with fractional damping, supported by computational algorithms, and a numerical example involving fractional diffusion in fractured media. The results provide meaningful refinements and novel insights that extend and enrich existing research in the field.

Open Access: Yes

DOI: 10.28924/2291-8639-24-2026-175

Some New Approaches of Hermite-Hadamard Type Inequalities Pertaining to Generalized Convexity on Coordinates Using Hypergeometric Function

Osama Oqilat Muhammad Tariq Hijaz Ahmad M. Nadeem Ali Asghar O. Oqilat Rakib Efendiev

Publication Name: Azerbaijan Journal of Mathematics

Publication Date: 2026-01-01

Volume: 16

Issue: 1

Page Range: 220-257

Description:

In this study, we develop a class of generalized fractional inequalities by employing (m, n)-polynomial (p1, p2)-convex functions defined on the coordinates. A novel integral identity for functions of two variables is established, serving as a key tool in our analysis. Furthermore, we derive new sort of Hermite-Hadamard-type inequality through generalized fractional operators. In addition, we present some new refinements of Hermite-Hadamard type inequality via (m, n)-polynomial (p1, p2)-convex functions with the help of hypergeometric functions. This framework provides a unified approach that encompasses several existing concepts, including (m, n)-polynomial harmonic convexity, (m, n)-polynomial convexity, classical harmonic convexity, and classical convexity, all obtained as specific instances of our results. Consequently, the findings presented here not only extend previously known inequalities but also recover a number of recent contributions in the literature as particular cases.

Open Access: Yes

DOI: 10.59849/2218-6816.2026.1.220

New Fractional Approach of Hermite-Hadamard-Type Inequalities with Applications to Information Divergence and Entropy Bounds

Evren Hincal Muhammad Tariq Hijaz Ahmad Tonguc Cagin Ilyas Khan Waleed Mohammed Abdelfattah Ali Asghar Waqar Afzal Saima Bhatti

Publication Name: International Journal of Analysis and Applications

Publication Date: 2026-01-01

Volume: 24

Issue: Unknown

Page Range: Unknown

Description:

Fractional integral operators play a vital role in establishing generalized forms of mathematical inequalities. These operators provide effective tools for modeling various scientific and engineering processes such as fracture mechanics, elasticity, heat transfer, viscoelastic deformation, and the behavior of continuous populations. In this study, we investigate a new class of Hermite–Hadamard type inequalities and verify their numerical validity. By employing a novel equality together with Hölder’s inequality, we derive several extensions of Hermite–Hadamard type inequalities through generalized convexity involving Raina’s function within the framework of fractional integral operators. Moreover, we present applications related to information divergence and entropy bounds. The results obtained here constitute significant advancements and generalizations of existing findings in the literature.

Open Access: Yes

DOI: 10.28924/2291-8639-24-2026-204

Boundedness Properties of Multidimensional Fractional Hadamard-Type Operators on Campanato Spaces

Evren Hincal Khurram Shabbir Mutum Zico Meetei Haitham Qawaqneh Muhammad Tariq Hijaz Ahmad Waqar Afzal Mohammed Abbas Ei-Naggar

Publication Name: International Journal of Analysis and Applications

Publication Date: 2026-01-01

Volume: 24

Issue: Unknown

Page Range: Unknown

Description:

The present work aims to extend the one-dimensional Hadamard fractional operator to a general multidimensional setting and study how this newly built operator acts between two Campanato spaces, Kp,µ (D) and Kq,ν (D). Alongside the boundedness statement itself, we pin down an explicit rate at which the operator norm grows with the diameter of D. Two features distinguish this study from prior boundedness results for fractional-type operators. First, the Campanato scale is strictly finer than the Morrey scale on which such questions are usually posed. Second, the Hadamard operator sits inside the Katugampola family as the degenerate case ρ → 0+, carrying a logarithmic kernel (log v σ)s−1 in place of the power kernels that dominate the classical theory – and it is exactly this logarithmic structure that drives the sharper estimates obtained below.

Open Access: Yes

DOI: 10.28924/2291-8639-24-2026-200

WOLFF POTENTIAL AND BIRKHOFF’S ERGODIC THEOREM IN GRAND VARIABLE HERZ SPACES

Rosalio G. Artes Muhammad Tariq Hijaz Ahmad Waqar Afzal Samra Asjad

Publication Name: Gulf Journal of Mathematics

Publication Date: 2026-07-20

Volume: 23

Issue: 2

Page Range: Unknown

Description:

In this article, we establish the boundedness of the Wolff potential operator and Birkhoff’s ergodic theorem in grand variable Herz spaces. These concepts were previously studied in the setting of Lebesgue spaces. Our results extend and refine the existing literature by considering a more general functional framework, thereby providing broader applicability and sharper estimates than the previously known results.

Open Access: Yes

DOI: 10.56947/51vxdc48

New Variant of Hermite-Hadamard-Type Inequality with Applications to Matrix and Bivariate Analysis

Evren Hincal Muhammad Tariq Hijaz Ahmad Ilyas Khan Waleed Mohammed Abdelfattah Ali Asghar Waqar Afzal Saima Bhatti Mustafa Bayram

Publication Name: International Journal of Analysis and Applications

Publication Date: 2026-01-01

Volume: 24

Issue: Unknown

Page Range: Unknown

Description:

Convex analysis and mathematical inequalities are now essential to the development of numerous pure and applied scientific domains. In this article, first we introduce the concept of n-fractional polynomial s-like m-preinvex function. In addition, we introduce, the new variant of Hermite-Hadamard type inequality via newly introduced concept pertaining to the k-fractional operator. Furthermore, we demonstrate the applications to matrix and bivariat analysis via newly introduce Hermite-Hadamard type inequality. The study’s conclusions present unique perspectives and contributions to the field, offering fresh and noteworthy improvements over previous research.

Open Access: Yes

DOI: 10.28924/2291-8639-24-2026-220

On Boundedness, Continuity, and Hyers–Ulam Stability of Interval-Valued Stochastic Processes in a Probabilistic Framework

Mujahid Abbas Muhammad Tariq Hijaz Ahmad Waleed Mohammed Abdelfattah Waqar Afzal

Publication Name: Statistics Optimization and Information Computing

Publication Date: 2026-08-03

Volume: 16

Issue: 3

Page Range: 2185-2199

Description:

In this paper, we investigate interval-valued stochastic processes within the framework of pseudo-order relations and analyze their fundamental analytical properties. We give a probabilistically meaningful definition of boundedness for interval-valued processes and use it to establish a single, consolidated local-to-global boundedness theorem, together with a corrected treatment of continuity in probability and of Hyers–Ulam stability for interval-valued stochastic mappings. Every construction that relies on interval subtraction, interval absolute value, or the ordering of negative intervals is given an explicit. The proposed approach provides a consistent structure for handling uncertainty arising from both randomness and interval-valued data, and we illustrate it with a toy financial example in which a return is simultaneously random and only known up to a measurement interval. We show precisely how the developed results relate to, and in the component-wise pseudo-order case reduce to, existing real-valued results, and we discuss open problems for the genuinely set-valued (inclusion-order) setting.

Open Access: Yes

DOI: 10.19139/soic-2310-5070-3884

Error Estimates for Numerical Quadrature Rules via New Fractional Integral Inequalities Based on the Prabhakar Fractional Operator for Harmonic Mappings with Computational and Graphical Validation

Waqas Nazeer Mutum Zico Meetei Mujahid Abbas Haitham Qawaqneh Lütfi Akın Mohamed Abbas El-Naggar Muhammad Tariq Hijaz Ahmad Waqar Afzal

Publication Name: International Journal of Analysis and Applications

Publication Date: 2026-01-01

Volume: 24

Issue: Unknown

Page Range: Unknown

Description:

In this work, we establish several new variants of the Hermite–Hadamard integral inequality, together with various product-type extensions, by employing the Prabhakar fractional integral operator with three-parameter Mittag-Leffler kernels for mappings defined on harmonic sets. This unified operator generalizes several classical fractional operators, including Riemann–Liouville, Erdélyi–Kober, and Weyl, which are recovered as special cases. The results obtained herein constitute natural generalizations and significant improvements of existing results developed using classical integral operators. To demonstrate the validity and effectiveness of the obtained results, we present nontrivial numerical examples supported by graphical illustrations and tabulated comparisons for different choices of the fractional parameters. Furthermore, as an application of our main results, we derive explicit error estimates for the trapezoidal rule on harmonic intervals.

Open Access: Yes

DOI: 10.28924/2291-8639-24-2026-242