Browsing by Author "Muhammad Aamir Ali"
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Item A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite-Hadamard-Type Inequalities with Applications(Walter de Gruyter GmbH, 2024) Muhammad Aamir Ali; Michal Fe_kan; Chanon Promsakon; Thanin Sitthiwirattham; M. Fe_kan; Department of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Bratislava, Mlynsk‡ dolina, 842 48, Slovakia; email: michal.feckan@fmph.uniba.skThe primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann-Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite-Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper. © 2024 Mathematical Institute Slovak Academy of Sciences.Item A NEW q-HERMITE-HADAMARD'S INEQUALITY AND ESTIMATES FOR MIDPOINT TYPE INEQUALITIES FOR CONVEX FUNCTIONS(University of Miskolc, 2023) Thanin Sitthiwirattham; Muhammad Aamir Ali; Asghar Ali; HŸseyin Budak; M.A. Ali; Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, 210023, China; email: mahr.muhammad.aamir@gmail.comThis paper proves a new q-Hermite-Hadamard inequality for convex functions using quantum integrals. We also prove some new midpoint-type inequalities for q-differentiable convex functions. Moreover, we present some examples to illustrate our established results, supplemented with graphs. © (2023) Miskolc University PressItem A new variant of Jensen inclusion and Hermite-Hadamard type inclusions for interval-valued functions(University of Nis, 2023) Thanin Sitthiwirattham; Ifra Bashir Sial; Muhammad Aamir Ali; HŸseyin Budak; Jiraporn Reunsumrit; M.A. Ali; Jiangsu Key Laboratory of NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.comIn this research, we give a new version of Jensen inclusion for interval-valued functions, which is called Jensen-Mercer inclusion. Moreover, we establish some new inclusions of the Hermite-Hadamard-Mercer type for interval-valued functions. Finally, we give some applications of newly established inequalities to make them more interesting for the readers. © 2023, University of Nis. All rights reserved.Item A new version of (p, q) -HermiteÐHadamardÕs midpoint and trapezoidal inequalities via special operators in (p, q) -calculus(Springer Science and Business Media Deutschland GmbH, 2022) Thanin Sitthiwirattham; Muhammad Aamir Ali; HŸseyin Budak; Sina Etemad; Shahram Rezapour; S. Rezapour; Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran; email: sh.rezapour@azaruniv.ac.irIn this paper, we conduct a research on a new version of the (p, q) -HermiteÐHadamard inequality for convex functions in the framework of postquantum calculus. Moreover, we derive several estimates for (p, q) -midpoint and (p, q) -trapezoidal inequalities for special (p, q) -differentiable functions by using the notions of left and right (p, q) -derivatives. Our newly obtained inequalities are extensions of some existing inequalities in other studies. Lastly, we consider some mathematical examples for some (p, q) -functions to confirm the correctness of newly established results. © 2022, The Author(s).Item A STUDY OF FRACTIONAL HERMITE-HADAMARD-MERCER INEQUALITIES FOR DIFFERENTIABLE FUNCTIONS(World Scientific, 2024) Thanin Sitthiwirattham; Miguel Vivas-Cortez; Muhammad Aamir Ali; HŸseyin Budak; Ibrahim Avci; M. Vivas-Cortez; School of Physical and Mathematical Sciences, Faculty of Exact and Natural Sciences Pontifical Catholic, University of Ecuador, Quito, Av. 12 October 1076, Section, 17-01-2184, Ecuador; email: mjvivas@puce.edu.ecIn this work, we prove a parameterized fractional integral identity involving differentiable functions. Then, we use the newly established identity to establish some new parameterized fractional Hermite-Hadamard-Mercer-type inequalities for differentiable function. The main benefit of the newly established inequalities is that these inequalities can be converted into some new Mercer inequalities of midpoint type, trapezoidal type, and Simpson's type for differentiable functions. Finally, we show the validation of the results with the help of some mathematical examples and their graphs. © The Author(s)Item FRACTIONAL HERMITE-HADAMARD INEQUALITY AND ERROR ESTIMATES FOR SIMPSONÕS FORMULA THROUGH CONVEXITY WITH RESPECT TO A PAIR OF FUNCTIONS(University of Miskolc, 2023) Muhammad Aamir Ali; Jarunee Soontharanon; HŸseyin Budak; Thanin Sitthiwirattham; Michal Fe_kan; M. Fe_kan; Department of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Bratislava, Mlynsk‡ dolina, 842 48, Slovakia; email: michal.feckan@fmph.uniba.skIn this article, we establish two new and different versions of fractional Hermite-Hadamard type inequality for the convex functions with respect to a pair of functions. Moreover, we establish a new SimpsonÕs type inequalities for differentiable convex functions with respect to a pair of functions. We also prove two more SimpsonÕs type inequalities for differentiable convex functions with respect to a pair of functions using the power mean and HšlderÕs inequalities. It is also shown that the newly established inequalities are the extension of some existing results. Finally, we add some mathematical examples and their graphs to show the validity of newly established results. © 2023 Miskolc University PressItem FRACTIONAL HERMITEÐHADAMARD INEQUALITY, SIMPSONÕS AND OSTROWSKIÕS TYPE INEQUALITIES FOR CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF FUNCTIONS(Rocky Mountain Mathematics Consortium, 2023) Jianqiang Xie; Muhammad Aamir Ali; HŸseyin Budak; Michal Fe_kan; Thanin Sitthiwirattham; M.A. Ali; School of Mathematical Sciences, Nanjing Normal University, Nanjing, China; email: mahr.muhammad.aamir@gmail.comWe consider the convexity with respect to a pair of functions and establish a HermiteÐHadamard type inequality for RiemannÐLiouville fractional integrals. Moreover, we derive some new SimpsonÕs and OstrowskiÕs type inequalities for differentiable convex mapping with respect to a pair of functions. We also show that the newly established inequalities are the extension of some existing inequalities. Finally, we consider some mathematical examples and graphs to show the validity of the newly established inequalities. © Rocky Mountain Mathematics Consortium.Item Fractional ostrowski type inequalities for differentiable harmonically convex functions(American Institute of Mathematical Sciences, 2022) Thanin Sitthiwirattham; Muhammad Aamir Ali; HŸseyin Budak; Sotiris K. Ntouyas; Chanon Promsakon; M.A. Ali; Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.comIn this paper, we prove some new Ostrowski type inequalities for differentiable harmonically convex functions using generalized fractional integrals. Since we are using generalized fractional integrals to establish these inequalities, therefore we obtain some new inequalities of Ostrowski type for Riemann-Liouville fractional integrals and k-Riemann-Liouville fractional integrals in special cases. Finally, we give some applications to special means of real numbers for newly established inequalities. © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0).Item HermiteÐhadamardÐmercer-type inequalities for harmonically convex mappings(MDPI, 2021) Xuexiao You; Muhammad Aamir Ali; HŸseyin Budak; Jiraporn Reunsumrit; Thanin Sitthiwirattham; M.A. Ali; Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.com; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand; email: thanin_sit@dusit.ac.thIn this paper, we prove HermiteÐHadamardÐMercer inequalities, which is a new version of the HermiteÐHadamard inequalities for harmonically convex functions. We also prove HermiteÐ HadamardÐMercer-type inequalities for functions whose first derivatives in absolute value are harmonically convex. Finally, we discuss how special means can be used to address newly discovered inequalities. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item Montgomery identity and Ostrowski-type inequalities via quantum calculus(De Gruyter Open Ltd, 2021) Thanin Sitthiwirattham; Muhammad Aamir Ali; Huseyin Budak; Mujahid Abbas; Saowaluck Chasreechai; M.A. Ali; Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.comIn this paper, we prove a quantum version of Montgomery identity and prove some new Ostrowski-type inequalities for convex functions in the setting of quantum calculus. Moreover, we discuss several special cases of newly established inequalities and obtain different new and existing inequalities in the field of integral inequalities. © 2021 Thanin Sitthiwirattham et al., published by De Gruyter.Item New Forms of the Open Newton-Cotes-Type Inequalities for a Family of the Quantum Differentiable Convex Functions(University of Maragheh, 2025) Jarunee Soontharanon; Muhammad Aamir Ali; Shahram Rezapour; Muhammad Toseef; Thanin Sitthiwirattham; S. Rezapour; Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran; email: rezapourshahram@yahoo.caThe main objective of this paper is to establish some new inequalities related to the open Newton-Cotes formulas in the setting of q-calculus. We establish a quantum integral identity first and then prove the desired inequalities for q-differentiable convex functions. These inequalities are useful for determining error bounds for the open Newton-Cotes formulas in both classical and q-calculus. This work distinguishes itself from existing studies by employing quantum operators, leading to sharper and more precise error estimates. These results extend the applicability of Newton-Cotes methods to quantum calculus, offering a novel contribution to the numerical analysis of convex functions. Finally, we provide mathematical examples and computational analysis to validate the newly established inequalities. © 2025 University of Maragheh. All rights reserved.Item On Generalization of Different Integral Inequalities for Harmonically Convex Functions(MDPI, 2022) Jiraporn Reunsumrit; Miguel J. Vivas-Cortez; Muhammad Aamir Ali; Thanin Sitthiwirattham; M.J. Vivas-Cortez; Escuela de Ciencias Matem‡ticas y F’sicas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Cat—lica del Ecuador, Quito, Av. 12 de Octubre 1076, Apartado, 17-01-2184, Ecuador; email: mjvivas@puce.edu.ecIn this study, we first prove a parameterized integral identity involving differentiable functions. Then, for differentiable harmonically convex functions, we use this result to establish some new inequalities of a midpoint type, trapezoidal type, and Simpson type. Analytic inequalities of this type, as well as the approaches for solving them, have applications in a variety of domains where symmetry is important. Finally, several particular cases of recently discovered results are discussed, as well as applications to the special means of real numbers. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.Item On generalizations of quantum SimpsonÕs and quantum newtonÕs inequalities with some parameters(American Institute of Mathematical Sciences, 2021) Chanon Promsakon; Muhammad Aamir Ali; HŸseyin Budak; Mujahid Abbas; Faheem Muhammad; Thanin Sitthiwirattham; M.A. Ali; Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.com; T. Sitthiwirattham; Department of Medical research, China Medical University Hospital, China, Medical University, Taichung, Taiwan; email: sit@dusit.ac.thIn this paper, we prove two identities concerning quantum derivatives, quantum integrals, and some parameters. Using the newly proved identities, we prove new SimpsonÕs and NewtonÕs type inequalities for quantum differentiable convex functions with two and three parameters, respectively. We also look at the special cases of our key findings and find some new and old SimpsonÕs type inequalities, NewtonÕs type inequalities, midpoint type inequalities, and trapezoidal type inequalities. © 2021 the Author(s), licensee AIMS Press.Item On New Estimates of q-HermiteÐHadamard Inequalities with Applications in Quantum Calculus(MDPI, 2023) Saowaluck Chasreechai; Muhammad Aamir Ali; Muhammad Amir Ashraf; Thanin Sitthiwirattham; Sina Etemad; Manuel De la Sen; Shahram Rezapour; S. Etemad; Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, 3751-71379, Iran; email: sina.etemad@azaruniv.ac.ir; S. Rezapour; Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, 3751-71379, Iran; email: sh.rezapour@azaruniv.ac.ir; M.D.L. Sen; Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV/EHU), Leioa, Bizkaia, 48940, Spain; email: manuel.delasen@ehu.eusIn this paper, we first establish two quantum integral (q-integral) identities with the help of derivatives and integrals of the quantum types. Then, we prove some new q-midpoint and q-trapezoidal estimates for the newly established q-Hermite-Hadamard inequality (involving left and right integrals proved by Bermudo et al.) under q-differentiable convex functions. Finally, we provide some examples to illustrate the validity of newly obtained quantum inequalities. © 2023 by the authors.Item On some error bounds of MaclaurinÕs formula for convex functions in q-calculus(University of Nis, 2023) Thanin Sitthiwirattham; Muhammad Aamir Ali; Jarunee Soontharanon; M.A. Ali; School of Mathematical Sciences, Nanjing Normal University, China; email: mahr.muhammad.aamir@gmail.comThe main goal of this paper is to establish some error bounds for MaclaurinÕs formula which is three point quadrature formula using the notions of q-calculus. For this, we first prove a q-integral identity involving fist time q-differentiable functions. Then, by using the new established identity we find the error bounds for maclaurinÕs formula by using the convexity of fist time q-differentiable functions. It is also shown that the newly established inequalities are extension of some existing inequalities inside the literature. © 2023, University of Nis. All rights reserved.Item On some new fractional ostrowski-and trapezoid-type inequalities for functions of bounded variations with two variables(MDPI, 2021) Thanin Sitthiwirattham; HŸseyin Budak; Hasan Kara; Muhammad Aamir Ali; Jiraporn Reunsumrit; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand; email: thanin_sit@dusit.ac.th; M.A. Ali; Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.comIn this paper, we first prove three identities for functions of bounded variations. Then, by using these equalities, we obtain several trapezoid-and Ostrowski-type inequalities via generalized fractional integrals for functions of bounded variations with two variables. Moreover, we present some results for RiemannÐLiouville fractional integrals by special choice of the main results. Finally, we investigate the connections between our results and those in earlier works. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications(De Gruyter Open Ltd, 2022) Xue-Xiao You; Muhammad Aamir Ali; Humaira Kalsoom; Jarunee Soontharanon; Thanin Sitthiwirattham; M.A. Ali; Jiangsu Key Laboratory for Nslscs, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.com; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand; email: thanin_sit@dusit.ac.thIn this study, we establish some new Hermite-Hadamard type inequalities for s-convex functions in the second sense using the post-quantum calculus. Moreover, we prove a new (p, q)\left(p,q)-integral identity to prove some new Ostrowski type inequalities for (p, q)\left(p,q)-differentiable functions. We also show that the newly discovered results are generalizations of comparable results in the literature. Finally, we give application to special means of real numbers using the newly proved inequalities. © 2022 Xue-Xiao You et al., published by De Gruyter.Item On some new inequalities of hermiteÐhadamard midpoint and trapezoid type for preinvex functions in (P, q)-calculus(MDPI, 2021) Ifra Bashir Sial; Muhammad Aamir Ali; Ghulam Murtaza; Sotiris K. Ntouyas; Jarunee Soontharanon; Thanin Sitthiwirattham; M.A. Ali; Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.com; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand; email: thanin_sit@dusit.ac.thIn this paper, we establish some new HermiteÐHadamard type inequalities for preinvex functions and left-right estimates of newly established inequalities for (p, q)-differentiable preinvex functions in the context of (p, q)-calculus. We also show that the results established in this paper are generalizations of comparable results in the literature of integral inequalities. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item On Some New MaclaurinÕs Type Inequalities for Convex Functions in q-Calculus(Multidisciplinary Digital Publishing Institute (MDPI), 2023) Thanin Sitthiwirattham; Muhammad Aamir Ali; HŸseyin Budak; H. Budak; Department of Mathematics, Faculty of Science and Arts, DŸzce University, DŸzce, 81620, Turkey; email: hsyn.budak@gmail.comThis work establishes some new inequalities to find error bounds for MaclaurinÕs formulas in the framework of q-calculus. For this, we first prove an integral identity involving q-integral and q-derivative. Then, we use this new identity to prove some q-integral inequalities for q-differentiable convex functions. The inequalities proved here are very important in the literature because, with their help, we can find error bounds for MaclaurinÕs formula in both q and classical calculus. © 2023 by the authors.Item On Some New OstrowskiÐMercer-Type Inequalities for Differentiable Functions(MDPI, 2022) Ifra Bashir Sial; Nichaphat Patanarapeelert; Muhammad Aamir Ali; HŸseyin Budak; Thanin Sitthiwirattham; M.A. Ali; Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China; email: mahr.muhammad.aamir@gmail.com; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand; email: thanin_sit@dusit.ac.thIn this paper, we establish a new integral identity involving differentiable functions, and then we use the newly established identity to prove some OstrowskiÐMercer-type inequalities for differentiable convex functions. It is also demonstrated that the newly established inequalities are generalizations of some of the Ostrowski inequalities established inside the literature. There are also some applications to the special means of real numbers given. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.