Hermite-Hadamard, Fejér and trapezoid type inequalities using Godunova-Levin Preinvex functions via Bhunia's order and with applications to quadrature formula and random variable

Waqar Afzal; Najla M. Aloraini; Mujahid Abbas; Jong-Suk Ro; Abdullah A. Zaagan; Waqar Afzal; Najla M. Aloraini; Mujahid Abbas; Jong-Suk Ro; Abdullah A. Zaagan

doi:10.3934/mbe.2024151

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 3422-3447. doi: 10.3934/mbe.2024151

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Hermite-Hadamard, Fejér and trapezoid type inequalities using Godunova-Levin Preinvex functions via Bhunia's order and with applications to quadrature formula and random variable

1.
Department of Mathematics, Government College University, Katchery Road, Lahore 54000, Pakistan
2.
Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung, Taiwan
4.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
5.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea
6.
Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Saudi Arabia

Received: 24 November 2023 Revised: 15 December 2023 Accepted: 27 December 2023 Published: 04 February 2024

Convex and preinvex functions are two different concepts. Specifically, preinvex functions are generalizations of convex functions. We created some intriguing examples to demonstrate how these classes differ from one another. We showed that Godunova-Levin invex sets are always convex but the converse is not always true. In this note, we present a new class of preinvex functions called $ (\mathtt{h_1}, \mathtt{h_2}) $-Godunova-Levin preinvex functions, which is extensions of $ \mathtt{h} $-Godunova-Levin preinvex functions defined by Adem Kilicman. By using these notions, we initially developed Hermite-Hadamard and Fejér type results. Next, we used trapezoid type results to connect our inequality to the well-known numerical quadrature trapezoidal type formula for finding error bounds by limiting to standard order relations. Additionally, we use the probability density function to relate trapezoid type results for random variable error bounds. In addition to these developed results, several non-trivial examples have been provided as proofs.
- Hermite–Hadamard,
- Fejer,
- Trapezoidal formula,
- Godunova-Levin preinvex,
- mathematical operators,
- random variable
Citation: Waqar Afzal, Najla M. Aloraini, Mujahid Abbas, Jong-Suk Ro, Abdullah A. Zaagan. Hermite-Hadamard, Fejér and trapezoid type inequalities using Godunova-Levin Preinvex functions via Bhunia's order and with applications to quadrature formula and random variable[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 3422-3447. doi: 10.3934/mbe.2024151

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Abstract

Convex and preinvex functions are two different concepts. Specifically, preinvex functions are generalizations of convex functions. We created some intriguing examples to demonstrate how these classes differ from one another. We showed that Godunova-Levin invex sets are always convex but the converse is not always true. In this note, we present a new class of preinvex functions called $ (\mathtt{h_1}, \mathtt{h_2}) $-Godunova-Levin preinvex functions, which is extensions of $ \mathtt{h} $-Godunova-Levin preinvex functions defined by Adem Kilicman. By using these notions, we initially developed Hermite-Hadamard and Fejér type results. Next, we used trapezoid type results to connect our inequality to the well-known numerical quadrature trapezoidal type formula for finding error bounds by limiting to standard order relations. Additionally, we use the probability density function to relate trapezoid type results for random variable error bounds. In addition to these developed results, several non-trivial examples have been provided as proofs.

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