On Hermite-Hadamard-type inequalities for second order differential inequalities with inverse-square potential

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

doi:10.3934/math.2024874

AIMS Mathematics

2024, Volume 9, Issue 7: 17955-17970. doi: 10.3934/math.2024874

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Research article

On Hermite-Hadamard-type inequalities for second order differential inequalities with inverse-square potential

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: 25 March 2024 Revised: 03 May 2024 Accepted: 16 May 2024 Published: 27 May 2024
MSC : 26A51, 26B25, 26D15

We consider the class of functions $ u\in C^2((0, \infty)) $ satisfying second-order differential inequalities in the form $ u''(x)+\frac{k}{x^2}u(x)\geq 0 $ for all $ x > 0 $. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases ($ k=\frac{1}{4} $ and $ 0 < k < \frac{1}{4} $). We next extend our obtained results to the two-dimensional case. In the limit case $ k\rightarrow 0^+ $ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations.
- Hermite-Hadamard-type inequalities,
- second order differential inequalities,
- convex functions,
- convex functions on the coordinates
Citation: Hassen Aydi, Bessem Samet, Manuel De la Sen. On Hermite-Hadamard-type inequalities for second order differential inequalities with inverse-square potential[J]. AIMS Mathematics, 2024, 9(7): 17955-17970. doi: 10.3934/math.2024874

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Abstract

We consider the class of functions $ u\in C^2((0, \infty)) $ satisfying second-order differential inequalities in the form $ u''(x)+\frac{k}{x^2}u(x)\geq 0 $ for all $ x > 0 $. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases ($ k=\frac{1}{4} $ and $ 0 < k < \frac{1}{4} $). We next extend our obtained results to the two-dimensional case. In the limit case $ k\rightarrow 0^+ $ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations.

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