Hermite-Hadamard and Ostrowski type inequalities in $ \mathfrak{h} $-calculus with applications

Miguel Vivas-Cortez; Muhammad Aamir Ali; Ghulam Murtaza; Ifra Bashir Sial; Miguel Vivas-Cortez; Muhammad Aamir Ali; Ghulam Murtaza; Ifra Bashir Sial

doi:10.3934/math.2022393

AIMS Mathematics

2022, Volume 7, Issue 4: 7056-7068. doi: 10.3934/math.2022393

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Research article

Hermite-Hadamard and Ostrowski type inequalities in $ \mathfrak{h} $-calculus with applications

1.
Pontificia Universidad Católica del Ecuador, Facultad de Ciencias Naturales y Exactas, Escuela de Ciencias Físicas y Matemáticas, Sede Quito, Ecuador
2.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China
3.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
4.
School of Sciences, Jiangsu University, Zhenjiang, China

Received: 12 August 2021 Revised: 30 November 2021 Accepted: 08 December 2021 Published: 08 February 2022
MSC : 26D07, 26D10, 26D15

In this paper, we prove Hermite-Hadamard inequality for convex functions in the framework of $ \mathfrak{h} $-calculus. We also use the notions of $ \mathfrak{h} $-derivative and $ \mathfrak{h} $-integral to prove Ostrowski's and trapezoidal type inequalities for bounded functions. It is also shown that the newly established inequalities are the generalization of the comparable inequalities in the literature. Finally, using some examples, we demonstrate the validity of newly formed inequalities and show how they can be used to special means of real numbers.
- Hermite-Hadamard inequality,
- Ostrowski inequality,
- $ \mathfrak{h} $ -integral,
- quantum calculus,
- convex functions
Citation: Miguel Vivas-Cortez, Muhammad Aamir Ali, Ghulam Murtaza, Ifra Bashir Sial. Hermite-Hadamard and Ostrowski type inequalities in $ \mathfrak{h} $-calculus with applications[J]. AIMS Mathematics, 2022, 7(4): 7056-7068. doi: 10.3934/math.2022393

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Abstract

In this paper, we prove Hermite-Hadamard inequality for convex functions in the framework of $ \mathfrak{h} $-calculus. We also use the notions of $ \mathfrak{h} $-derivative and $ \mathfrak{h} $-integral to prove Ostrowski's and trapezoidal type inequalities for bounded functions. It is also shown that the newly established inequalities are the generalization of the comparable inequalities in the literature. Finally, using some examples, we demonstrate the validity of newly formed inequalities and show how they can be used to special means of real numbers.

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