Fractional $ 3/8 $-Simpson type inequalities for differentiable convex functions

Nassima Nasri; Badreddine Meftah; Abdelkader Moumen; Hicham Saber; Nassima Nasri; Badreddine Meftah; Abdelkader Moumen; Hicham Saber

doi:10.3934/math.2024258

AIMS Mathematics

2024, Volume 9, Issue 3: 5349-5375. doi: 10.3934/math.2024258

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Fractional $ 3/8 $-Simpson type inequalities for differentiable convex functions

1.
Université 20 août 1955 Skikda Bp 26 Route El-Hadaiek 21000 Skikda, Algeria
2.
Laboratory of Analysis and Control of Differential Equations "ACED", Faculty MISM, Department of Mathematics, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria
3.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 55473, Saudi Arabia

Received: 10 December 2023 Revised: 22 January 2024 Accepted: 23 January 2024 Published: 26 January 2024
MSC : 26A51, 26D10, 26D15

The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases $ m = 1/2 $ and $ m = 3/8 $, as well as the two cases $ s = 1 $ and $ \alpha = 1 $, which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables.
- Riemann-Liouville integral operators,
- $ 3/8 $-Simpson inequality,
- convex functions,
- Hölder inequality
Citation: Nassima Nasri, Badreddine Meftah, Abdelkader Moumen, Hicham Saber. Fractional $ 3/8 $-Simpson type inequalities for differentiable convex functions[J]. AIMS Mathematics, 2024, 9(3): 5349-5375. doi: 10.3934/math.2024258

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Abstract

The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases $ m = 1/2 $ and $ m = 3/8 $, as well as the two cases $ s = 1 $ and $ \alpha = 1 $, which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables.

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