On $ \psi $-convex functions and related inequalities

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

doi:10.3934/math.2024546

AIMS Mathematics

2024, Volume 9, Issue 5: 11139-11155. doi: 10.3934/math.2024546

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On $ \psi $-convex functions and related inequalities

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: 09 March 2024 Accepted: 11 March 2024 Published: 21 March 2024
MSC : 26A51, 26D15, 26D10

We introduce the class of $ \psi $-convex functions $ f:[0, \infty)\to \mathbb{R} $, where $ \psi\in C([0, 1]) $ satisfies $ \psi\geq 0 $ and $ \psi(0)\neq \psi(1) $. This class includes several types of convex functions introduced in previous works. We first study some properties of such functions. Next, we establish a double Hermite-Hadamard-type inequality involving $ \psi $-convex functions and a Simpson-type inequality for functions $ f\in C^1([0, \infty)) $ such that $ |f'| $ is $ \psi $-convex. Our obtained results are new and recover several existing results from the literature.
- $ \psi $-convex functions,
- Hermite-Hadamard-type inequalities,
- Simpson-type inequalities
Citation: Hassen Aydi, Bessem Samet, Manuel De la Sen. On $ \psi $-convex functions and related inequalities[J]. AIMS Mathematics, 2024, 9(5): 11139-11155. doi: 10.3934/math.2024546

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Abstract

We introduce the class of $ \psi $-convex functions $ f:[0, \infty)\to \mathbb{R} $, where $ \psi\in C([0, 1]) $ satisfies $ \psi\geq 0 $ and $ \psi(0)\neq \psi(1) $. This class includes several types of convex functions introduced in previous works. We first study some properties of such functions. Next, we establish a double Hermite-Hadamard-type inequality involving $ \psi $-convex functions and a Simpson-type inequality for functions $ f\in C^1([0, \infty)) $ such that $ |f'| $ is $ \psi $-convex. Our obtained results are new and recover several existing results from the literature.

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