A generalization of convexity via an implicit inequality

Hassen Aydi; Bessem Samet; Manuel De la Sen; Hassen Aydi; Bessem Samet; Manuel De la Sen

doi:10.3934/math.2024586

AIMS Mathematics

2024, Volume 9, Issue 5: 11992-12010. doi: 10.3934/math.2024586

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A generalization of convexity via an implicit inequality

1.
Université de Sousse, Institut Supérieur d'Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
2.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa
3.
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
4.
Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, 48940-Leioa (Bizkaia), Spain

Received: 08 February 2024 Revised: 16 March 2024 Accepted: 19 March 2024 Published: 27 March 2024
MSC : 26A51, 26D15, 26D10

We unified several kinds of convexity by introducing the class $ \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ of $ (\zeta, w) $-admissible functions $ F: [0, 1]\times I\times I\to \mathbb{R} $. Namely, we proved that most types of convexity from the literature generate functions $ F\in \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ for some $ \zeta\in C([0, 1]) $ and $ w\in C^1(I) $ with $ w(I)\subset I $ and $ w' > 0 $. We also studied some properties of $ (\zeta, w) $-admissible functions and established some integral inequalities that unify various Hermite-Hadamard-type inequalities from the literature.
- convexity,
- implicit inequality,
- $ (\zeta, w) $-admissible functions,
- integral inequalities,
- Hermite-Hadamard-type inequalities
Citation: Hassen Aydi, Bessem Samet, Manuel De la Sen. A generalization of convexity via an implicit inequality[J]. AIMS Mathematics, 2024, 9(5): 11992-12010. doi: 10.3934/math.2024586

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Abstract

We unified several kinds of convexity by introducing the class $ \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ of $ (\zeta, w) $-admissible functions $ F: [0, 1]\times I\times I\to \mathbb{R} $. Namely, we proved that most types of convexity from the literature generate functions $ F\in \mathcal{A}_{\zeta, w}([0, 1]\times I^2) $ for some $ \zeta\in C([0, 1]) $ and $ w\in C^1(I) $ with $ w(I)\subset I $ and $ w' > 0 $. We also studied some properties of $ (\zeta, w) $-admissible functions and established some integral inequalities that unify various Hermite-Hadamard-type inequalities from the literature.

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