Mirajul Haq
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Mirajul Haq has a total of 23 co-authors. Here are the top co-authors by number of shared publications:
- Muhammad Arif
Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
5 shared publications - Bilal Khan
Institute of Mathematics, Henan Academy of Sciences, NO. 228, Chongshi Village, Zhengdong New District, Zhengzhou 450046, China
3 shared publications - Khurshid Ahmad
Department of Mathematics, Govt Post Graduate College, Dargai Malakand, Pakistan
3 shared publications - Nazar Khan
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan
3 shared publications - Abid Khan
School of Artificial Intelligence, Taizhou University, Taizhou 318000, China
2 shared publications - Ashfaq Ahmad Shah
College of Humanities and Development Studies (COHD), China Agricultural University, Beijing, China
2 shared publications
Latest Publications
- Open Access
- 14 October 2026
- Published by MDPI AG in Fractal and Fractional
- Vol. 10 (10) , 720
Fractional q-Starlikeness in Mittag–Leffler and Limaçon Domains
In this paper, we use a normalized fractional q-derivative operator, constructed by combining Jackson’s q-difference operator with a fractional q-integral, to introduce two subclasses of normalized analytic functions in the open unit disk whose fractional transforms are starlike and univalent. The first subclass, , is defined by subordination to a normalized q-Mittag–Leffler function, whereas the second, , is associated with a generalized limaçon domain. For both classes, we investigate sharp initial coefficient bounds and sharp estimates for the Fekete–Szegö functional. We also study inclusion and radius relations between the two classes. This comparative approach clarifies how an analytically defined Mittag–Leffler domain and a geometrically defined limaçon domain affect coefficient estimates when the underlying operator is held fixed. Relevant classical and quantum calculus results are recovered through suitable parameter choices and in the limiting case .- 14 October 2026
- Published by MDPI AG in Fractal and Fractional
- Vol. 10 (10) , 720
Open AccessFractional q-Starlikeness in Mittag–Leffler and Limaçon Domains
In this paper, we use a normalized fractional q-derivative operator, constructed by combining Jackson’s q-difference operator with a fractional q-integral, to introduce two subclasses of normalized analytic functions in the open unit disk whose fractional transforms are starlike and univalent. The first subclass, , is defined by subordination to a normalized q-Mittag–Leffler function, whereas the second, , is associated with a generalized limaçon domain. For both classes, we investigate sharp initial coefficient bounds and sharp estimates for the Fekete–Szegö functional. We also study inclusion and radius relations between the two classes. This comparative approach clarifies how an analytically defined Mittag–Leffler domain and a geometrically defined limaçon domain affect coefficient estimates when the underlying operator is held fixed. Relevant classical and quantum calculus results are recovered through suitable parameter choices and in the limiting case . - Open Access
- 2 July 2025
- Published by Wisdom Education & Research Hub in Physical Education, Health and Social Sciences
- Vol. 3 (3) , 71-79
A Study On Coefficient Bounds for a Newly Defined Subfamily of Convex Functions
One of the most crucial problems in geometric function theory is the study of the Hankel determinant generated by the Maclaurin series of analytic functions that belong to particular classes of normalized univalent functions. Our goal in this study is first to define a family of convex functions associated with Zigzag coefficients and then to investigate bounds of initial coefficients, Fekete-Szegö inequality, second and third-order Hankel determinants. Further, we also examine the logarithmic coefficients of functions within a defined family regarding recent issues.Mathematics Subject Classification.30C45, 30C50.
- 2 July 2025
- Published by Wisdom Education & Research Hub in Physical Education, Health and Social Sciences
- Vol. 3 (3) , 71-79
Open AccessA Study On Coefficient Bounds for a Newly Defined Subfamily of Convex Functions
One of the most crucial problems in geometric function theory is the study of the Hankel determinant generated by the Maclaurin series of analytic functions that belong to particular classes of normalized univalent functions. Our goal in this study is first to define a family of convex functions associated with Zigzag coefficients and then to investigate bounds of initial coefficients, Fekete-Szegö inequality, second and third-order Hankel determinants. Further, we also examine the logarithmic coefficients of functions within a defined family regarding recent issues.Mathematics Subject Classification.30C45, 30C50.
- 16 March 2023
- Published by Springer Nature in Environmental Science and Pollution Research
- Vol. 30 (19) , 56786-56801
A multi-criteria decision-making approach to vulnerability assessment of rural flooding in Khyber Pakhtunkhwa Province, Pakistan
Assessment of rural regions’ vulnerability to flooding is gaining prominence on a global scale. However, researchers are greatly undermined in their efforts to make a comprehensive assessment owing to the multidimensional and non-linear link between different indicators and flood risk. Thus, a multi-criteria decision-making (MCDM) approach is proposed to assess the multifaceted vulnerability of rural flooding in Khyber Pakhtunkhwa Province, Pakistan. This research presents a hybrid model for flood vulnerability assessment by combining TOPSIS and the entropy weight method. Households’ vulnerability to flooding in rural areas is assessed through four components (social, economic, physical, and institutional) and twenty indicators. All indicator weights are derived using the entropy weight method. The TOPSIS method is then used to rank the selected research areas based on their flood vulnerability levels. The ranking results reveal that flood vulnerability is highest in the Nowshehra District, followed by the Charsadda, Peshawar, and D.I. Khan Districts. The weighting results show that physical vulnerability is the most important component, while location of household’s house from the river source (< 1 km) is the key indicator for assessing flood vulnerability. A sensitivity analysis is provided to study the impact of indicator’s weights on the comprehensive ranking results. The sensitivity results revealed that out of twenty indicators, fourteen indicators had the lowest sensitivity, three indicators were reported with low sensitivity while the other three were considered highly sensitive for flood vulnerability assessment. Our research has the potential to offer policymakers specific guidelines for lowering flood risk in flood-prone areas.
- 16 March 2023
- Published by Springer Nature in Environmental Science and Pollution Research
- Vol. 30 (19) , 56786-56801
A multi-criteria decision-making approach to vulnerability assessment of rural flooding in Khyber Pakhtunkhwa Province, Pakistan
Assessment of rural regions’ vulnerability to flooding is gaining prominence on a global scale. However, researchers are greatly undermined in their efforts to make a comprehensive assessment owing to the multidimensional and non-linear link between different indicators and flood risk. Thus, a multi-criteria decision-making (MCDM) approach is proposed to assess the multifaceted vulnerability of rural flooding in Khyber Pakhtunkhwa Province, Pakistan. This research presents a hybrid model for flood vulnerability assessment by combining TOPSIS and the entropy weight method. Households’ vulnerability to flooding in rural areas is assessed through four components (social, economic, physical, and institutional) and twenty indicators. All indicator weights are derived using the entropy weight method. The TOPSIS method is then used to rank the selected research areas based on their flood vulnerability levels. The ranking results reveal that flood vulnerability is highest in the Nowshehra District, followed by the Charsadda, Peshawar, and D.I. Khan Districts. The weighting results show that physical vulnerability is the most important component, while location of household’s house from the river source (< 1 km) is the key indicator for assessing flood vulnerability. A sensitivity analysis is provided to study the impact of indicator’s weights on the comprehensive ranking results. The sensitivity results revealed that out of twenty indicators, fourteen indicators had the lowest sensitivity, three indicators were reported with low sensitivity while the other three were considered highly sensitive for flood vulnerability assessment. Our research has the potential to offer policymakers specific guidelines for lowering flood risk in flood-prone areas.
- Open Access
Majorization Results Based upon the Bernardi Integral Operator
By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, we investigate several majorization results for the class of normalized starlike functions, which are associated with the Janowski functions. We also give some particular cases of our main results. Finally, we direct the interested readers to the possibility of examining the fundamental or quantum (or -) extensions of the findings provided in this work in the concluding section. However, the -variations of the suggested -results will provide relatively minor and inconsequential developments because the additional (rather forced-in) parameter is obviously redundant.Open AccessMajorization Results Based upon the Bernardi Integral Operator
By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, we investigate several majorization results for the class of normalized starlike functions, which are associated with the Janowski functions. We also give some particular cases of our main results. Finally, we direct the interested readers to the possibility of examining the fundamental or quantum (or -) extensions of the findings provided in this work in the concluding section. However, the -variations of the suggested -results will provide relatively minor and inconsequential developments because the additional (rather forced-in) parameter is obviously redundant. - Open Access
- 13 April 2022
- Published by MDPI AG in Mathematics
- Vol. 10 (8) , 1297
A New Approach for Normal Parameter Reduction Using σ-Algebraic Soft Sets and Its Application in Multi-Attribute Decision Making
The soft set is one of the key mathematical tools for uncertainty description and has many applications in real-world decision-making problems. However, most of the time, these decision-making problems involve less important and redundant parameters, which make the decision making process more complex and challenging. Parameter reduction is a useful approach to eliminate such irrelevant and redundant parameters during soft set-based decision-making problems without changing their decision abilities. Among the various reduction methods of soft sets, normal parameter reduction (NPR) can reduce decision-making problems without changing the decision order of alternatives. This paper mainly develops a new algorithm for NPR using the concept of -algebraic soft sets. Before this, the same concept was used to introduce the idea of intersectional reduced soft sets (IRSSs). However, this study clarifies that the method of IRSSs does not maintain the decision order of alternatives. Thus, we need to develop a new approach that not only keeps the decision order invariant but also makes the reduction process more simple and convenient. For this reason, we propose a new algorithm for NPR using -algebraic soft sets that not only overcome the existing problems of IRSSs method but also reduce the computational complexity of the NPR process. We also compare our proposed algorithm with one of the existing algorithms of the NPR in terms of computational complexity. It is evident from the experimental results that the proposed algorithm has greatly reduced the computational complexity and workload in comparison with the existing algorithm. At the end of the paper, an application of the proposed algorithm is explored by a real-world decision-making problem.- 13 April 2022
- Published by MDPI AG in Mathematics
- Vol. 10 (8) , 1297
Open AccessA New Approach for Normal Parameter Reduction Using σ-Algebraic Soft Sets and Its Application in Multi-Attribute Decision Making
The soft set is one of the key mathematical tools for uncertainty description and has many applications in real-world decision-making problems. However, most of the time, these decision-making problems involve less important and redundant parameters, which make the decision making process more complex and challenging. Parameter reduction is a useful approach to eliminate such irrelevant and redundant parameters during soft set-based decision-making problems without changing their decision abilities. Among the various reduction methods of soft sets, normal parameter reduction (NPR) can reduce decision-making problems without changing the decision order of alternatives. This paper mainly develops a new algorithm for NPR using the concept of -algebraic soft sets. Before this, the same concept was used to introduce the idea of intersectional reduced soft sets (IRSSs). However, this study clarifies that the method of IRSSs does not maintain the decision order of alternatives. Thus, we need to develop a new approach that not only keeps the decision order invariant but also makes the reduction process more simple and convenient. For this reason, we propose a new algorithm for NPR using -algebraic soft sets that not only overcome the existing problems of IRSSs method but also reduce the computational complexity of the NPR process. We also compare our proposed algorithm with one of the existing algorithms of the NPR in terms of computational complexity. It is evident from the experimental results that the proposed algorithm has greatly reduced the computational complexity and workload in comparison with the existing algorithm. At the end of the paper, an application of the proposed algorithm is explored by a real-world decision-making problem. - Open Access
Fourth Hankel Determinant Problem Based on Certain Analytic Functions
In recent years, the Hankel determinant bounds for different subclasses of analytic, starlike and symmetric starlike functions have been discussed and studied by the many well-known authors. In this paper, we first consider a new subclass of analytic function and then we derive the fourth Hankel determinant bound for this class.
Open AccessFourth Hankel Determinant Problem Based on Certain Analytic Functions
In recent years, the Hankel determinant bounds for different subclasses of analytic, starlike and symmetric starlike functions have been discussed and studied by the many well-known authors. In this paper, we first consider a new subclass of analytic function and then we derive the fourth Hankel determinant bound for this class.
- Open Access
Radius of Star-Likeness for Certain Subclasses of Analytic Functions
In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition , for some analytic function, g, with ∀ We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp.Open AccessRadius of Star-Likeness for Certain Subclasses of Analytic Functions
In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition , for some analytic function, g, with ∀ We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp. - 1 January 2020
- Published by Mathematical Notes in Miskolc Mathematical Notes
- Vol. 21 (2) , 575-592
RELATIONSHIP OF THE BERNARDI INTEGRAL OPERATOR WITH CERTAIN CLASSES OF ANALYTIC FUNCTIONS
- 1 January 2020
- Published by Mathematical Notes in Miskolc Mathematical Notes
- Vol. 21 (2) , 575-592
RELATIONSHIP OF THE BERNARDI INTEGRAL OPERATOR WITH CERTAIN CLASSES OF ANALYTIC FUNCTIONS