Schur-Convexity of the Mean of Convex Functions for Two Variables
Abstract
:1. Introduction
2. Definitions and Lemmas
- (1)
- A set is said to be convex if and implies
- (2)
- Let be convex set. A function ψ: is said to be a convex function on Ω if, for all and all , inequality
- (1)
- is said to be majorized by (in symbols ) if and .
- (2)
- ψ is said to be a Schur-convex function on Ω if on implies , and ψ is said to be a Schur-concave function on Ω iff is a Schur-convex function.
3. Proofs of Main Results
4. Application on Binary Mean
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Shi, H.-N.; Wang, D.-S.; Fu, C.-R. Schur-Convexity of the Mean of Convex Functions for Two Variables. Axioms 2022, 11, 681. https://doi.org/10.3390/axioms11120681
Shi H-N, Wang D-S, Fu C-R. Schur-Convexity of the Mean of Convex Functions for Two Variables. Axioms. 2022; 11(12):681. https://doi.org/10.3390/axioms11120681
Chicago/Turabian StyleShi, Huan-Nan, Dong-Sheng Wang, and Chun-Ru Fu. 2022. "Schur-Convexity of the Mean of Convex Functions for Two Variables" Axioms 11, no. 12: 681. https://doi.org/10.3390/axioms11120681