Further improvements of the Jensen inequality in the integral sense by virtue of 6-convexity along with applications

Asadullah Sohail; Muhammad Adil Khan; Emad Abouel Nasr; Xiaoye Ding; Asadullah Sohail; Muhammad Adil Khan; Emad Abouel Nasr; Xiaoye Ding

doi:10.3934/math.2024553

AIMS Mathematics

2024, Volume 9, Issue 5: 11278-11303. doi: 10.3934/math.2024553

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Research article

Further improvements of the Jensen inequality in the integral sense by virtue of 6-convexity along with applications

1.
Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2.
Industrial Engineering Department, College of Engineering, King Saud University, Riyadh 11421, Saudi Arabia
3.
General Education Department, Anhui Xinhua University, Hefei 230088, China

Received: 25 January 2024 Revised: 06 March 2024 Accepted: 15 March 2024 Published: 22 March 2024
MSC : 39B62, 26D15, 94A15

The Jensen inequality is of fundamental importance because of its influential and interesting consequences. In recent years, the Jensen inequality has been supposed to be the most engaging source for research. We present interesting improvements to the continuous version of Jensen's inequality through the application of the concept of 6-convexity. For real visualization and comparison to other results, some numerical experiments were provided. With the aid of the acquired results, improvements for the Hermite-Hadamard and Hölder inequalities were presented. Some relationships between the means were granted as applications of established improvements. In addition, some estimations of the Csiszár divergence and its associated cases were received as further applications of the obtained results. The major techniques employed in formulating the proposed improvements included the Jensen inequality and the concept of convexity.
- information theory,
- convex function,
- Jensen's inequality,
- 6-convex function,
- power means,
- Hölder inequality,
- quasi-arithmetic means
Citation: Asadullah Sohail, Muhammad Adil Khan, Emad Abouel Nasr, Xiaoye Ding. Further improvements of the Jensen inequality in the integral sense by virtue of 6-convexity along with applications[J]. AIMS Mathematics, 2024, 9(5): 11278-11303. doi: 10.3934/math.2024553

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Abstract

The Jensen inequality is of fundamental importance because of its influential and interesting consequences. In recent years, the Jensen inequality has been supposed to be the most engaging source for research. We present interesting improvements to the continuous version of Jensen's inequality through the application of the concept of 6-convexity. For real visualization and comparison to other results, some numerical experiments were provided. With the aid of the acquired results, improvements for the Hermite-Hadamard and Hölder inequalities were presented. Some relationships between the means were granted as applications of established improvements. In addition, some estimations of the Csiszár divergence and its associated cases were received as further applications of the obtained results. The major techniques employed in formulating the proposed improvements included the Jensen inequality and the concept of convexity.

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