Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 41, 24 June 2024
Open
Access | Article
The application of convex function and GA-convex function
* Author to whom correspondence should be addressed.
Abstract
A convex function is a function that maps from a convex subset of a vector space to the set of real numbers. Convex functions have some important properties, such as non-negativity, monotonicity, and convexity, which can help us derive and prove inequalities. This paper explores the concepts of convex functions and GA-convex functions, demonstrating their utility in proving a variety of common and complex inequalities. Beginning with an overview of convex functions and their extension to GA-convex functions, the study shows how these mathematical tools can be effectively utilized in the context of inequality proofs. By leveraging the properties of these functions, the paper successfully establishes rigorous proofs for a range of inequalities, highlighting the versatility and applicability of convex and GA-convex functions in mathematical analysis. The properties convex and GA-convex functions allow us to use it to determine the direction of inequalities, prove inequalities, determine the optimal solution of inequalities, and even prove Cauchy inequalities.
Keywords
Convex function, GA-convex function, Application
References
1. Cha, L. (2004) Convex functions and inequalities. Journal of Ningbo Vocational and Technical College, 8, 3.
2. Xia, H. (2005). Convex functions and inequalities. Journal of Changzhou Institute of Technology, 18, 3.
3. Wu, S. (2005). Square convex functions and Jensen-type inequalities. Journal of Capital Normal University: Natural Science Edition, 26, 6.
4. Song, Z. and Wan, X. (2010). Hadamard-type inequalities for Ga-convex functions. Science, Technology and Engineering, 23, 3.
5. Shi, T., Wu, H. and Jiao, Z. (2013). Two functions related to Hermite-Hadamard type inequalities for Ga-convex functions. Journal of Guizhou Normal University: Natural Science Edition, 31, 5.
6. Shi, T. and Wu, H. (2013). Weighted Hadamard-type inequalities for differentiable Ga-convex functions. Journal of Chongqing University of Science and Technology: Natural Science, 6, 5.
7. Wu, Q. and Mao, Y. (2022). Properties of Multivariate Convex Functions and Their Hermite-Hadamard Inequality. Mathematics in Practice and Understanding, 52, 268-272.
8. Zhou, Z. (2006). In the process of proving inequalities, one must follow the general rules and basic methods of reasoning for proving problems, and also, due to the ‘inequality’ aspect, it is necessary to adopt some special proof methods. This article will use one of the properties of functions - convexity - to prove some inequalities in high school algebra. Journal of Lanzhou Institute of Education, 4, 58-60.
9. Wu, S. (2004). Ga-convex functions and the Poincaré-type inequality. Journal of Guizhou Normal University (Natural Science Edition), 2, 52-55.
10. Hua, Y. (2008). Hadamard-type inequalities for Ga-convex functions. College Mathematics, 24, 3.
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. Authors who publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See Open Access Instruction).
- Volume Title
- Proceedings of Machine Learning: Integrating machine learning techniques to advance network security - CONFMPCS 2024
- ISBN (Print)
- 978-1-83558-493-4
- ISBN (Online)
- 978-1-83558-494-1
- Published Date
- 24 June 2024
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/41/20240107
- Copyright
- 24 June 2024
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited