Unified inequalities of the $ {\mathfrak{q}} $-Trapezium-Jensen-Mercer type that incorporate majorization theory with applications

Bandar Bin-Mohsin; Muhammad Zakria Javed; Muhammad Uzair Awan; Hüseyin Budak; Awais Gul Khan; Clemente Cesarano; Muhammad Aslam Noor; Bandar Bin-Mohsin; Muhammad Zakria Javed; Muhammad Uzair Awan; Hüseyin Budak; Awais Gul Khan; Clemente Cesarano; Muhammad Aslam Noor

doi:10.3934/math.20231062

AIMS Mathematics

2023, Volume 8, Issue 9: 20841-20870. doi: 10.3934/math.20231062

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Unified inequalities of the $ {\mathfrak{q}} $-Trapezium-Jensen-Mercer type that incorporate majorization theory with applications

1.
Department of Mathematics, College of Science King Saud University, Riyadh, Saudi Arabia
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey
4.
Section of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Roma, Italy
5.
Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

Received: 15 March 2023 Revised: 04 May 2023 Accepted: 09 May 2023 Published: 29 June 2023
MSC : 05A30, 26A51, 26D10, 26D15

The objective of this paper is to explore novel unified continuous and discrete versions of the Trapezium-Jensen-Mercer (TJM) inequality, incorporating the concept of convex mapping within the framework of $ {\mathfrak{q}} $-calculus, and utilizing majorized tuples as a tool. To accomplish this goal, we establish two fundamental lemmas that utilize the $ _{{\varsigma_{1}}}{\mathfrak{q}} $ and $ ^{{{\varsigma_{2}}}}{\mathfrak{q}} $ differentiability of mappings, which are critical in obtaining new left and right side estimations of the midpoint $ {\mathfrak{q}} $-TJM inequality in conjunction with convex mappings. Our findings are significant in a way that they unify and improve upon existing results. We provide evidence of the validity and comprehensibility of our outcomes by presenting various applications to means, numerical examples, and graphical illustrations.
- convex,
- quantum,
- trapezium,
- Jensen-Mercer,
- differentiable,
- majorization
Citation: Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan, Hüseyin Budak, Awais Gul Khan, Clemente Cesarano, Muhammad Aslam Noor. Unified inequalities of the $ {\mathfrak{q}} $-Trapezium-Jensen-Mercer type that incorporate majorization theory with applications[J]. AIMS Mathematics, 2023, 8(9): 20841-20870. doi: 10.3934/math.20231062

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Abstract

The objective of this paper is to explore novel unified continuous and discrete versions of the Trapezium-Jensen-Mercer (TJM) inequality, incorporating the concept of convex mapping within the framework of $ {\mathfrak{q}} $-calculus, and utilizing majorized tuples as a tool. To accomplish this goal, we establish two fundamental lemmas that utilize the $ _{{\varsigma_{1}}}{\mathfrak{q}} $ and $ ^{{{\varsigma_{2}}}}{\mathfrak{q}} $ differentiability of mappings, which are critical in obtaining new left and right side estimations of the midpoint $ {\mathfrak{q}} $-TJM inequality in conjunction with convex mappings. Our findings are significant in a way that they unify and improve upon existing results. We provide evidence of the validity and comprehensibility of our outcomes by presenting various applications to means, numerical examples, and graphical illustrations.

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