Browsing by Author "Ghulam Murtaza"

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    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.th
    In 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.
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    On some new trapezoidal type inequalities for twice (P, q) differentiable convex functions in post-quantum calculus
    (MDPI, 2021) Thanin Sitthiwirattham; Ghulam Murtaza; Muhammad Aamir Ali; Sotiris K. Ntouyas; Muhammad Adeel; Jarunee Soontharanon; T. Sitthiwirattham; Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand; email: thanin_sit@dusit.ac.th
    Quantum information theory, an interdisciplinary field that includes computer science, information theory, philosophy, cryptography, and symmetry, has various applications for quantum calculus. Inequalities has a strong association with convex and symmetric convex functions. In this study, first we establish a (p,q)-integral identity involving the second (p,q)-derivative and then we used this result to prove some new trapezoidal type inequalities for twice (p,q)-differentiable convex functions. It is also shown that the newly established results are the refinements of some existing results in the field 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.
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    Post-Quantum Midpoint-Type Inequalities Associated with Twice-Differentiable Functions
    (MDPI, 2022) Thanin Sitthiwirattham; Ghulam Murtaza; Muhammad Aamir Ali; Chanon Promsakon; Ifra Bashir Sial; Praveen Agarwal; 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.com
    In this study, first we establish a (p, q)-integral identity involving the second (p, q)-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-(p, q)-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature. © 2022 by the authors. Licensee MDPI, Basel, Switzerland.
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    Post-quantum simpsonÕs type inequalities for coordinated convex functions
    (American Institute of Mathematical Sciences, 2022) Xue-Xiao You; Muhammad Aamir Ali; Ghulam Murtaza; Saowaluck Chasreechai; Sotiris K. Ntouyas; 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.th
    In this paper, we prove some new SimpsonÕs type inequalities for partial (p, q)-differentiable convex functions of two variables in the context of (p, q)-calculus. We also show that the findings in this paper are generalizations of comparable findings in the literature. © 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).