Schur-Power Convexity of a Completely Symmetric Function Dual
Abstract
:1. Introduction and Preliminaries
- (i)
- A set is called symmetric, if implies for every permutation matrix P.
- (ii)
- A function is called symmetric if for every permutation matrix P, for all .
- (i)
- means for all .
- (ii)
- Let , φ: is said to be increasing if implies . φ is said to be decreasing if and only if is increasing.
- (i)
- is said to be majorized by (in symbols ) if for and , where and are rearrangements of and in a descending order.
- (ii)
- Let , the function φ: is said to be Schur-convex on Ω if on implies φ is said to be a Schur-concave function on Ω if and only if is a Schur-convex function on Ω.
- (i)
- A set is called a geometrically convex set if for all , and α, such that .
- (ii)
- Let . The function is said to be Schur-geometrically convex on Ω if on Ω implies . The function φ is said to be a Schur-geometrically concave on Ω if and only if is Schur-geometrically convex on Ω.
- (i)
- A set Ω is said to be harmonically convex if for every and , where and .
- (ii)
- A function is said to be Schur-harmonically convex on Ω if implies . A function φ is said to be a Schur-harmonically concave function on Ω if and only if is a Schur-harmonically convex function.
- (i)
- is increasing and Schur-convex on ;
- (ii)
- if r is even integer (or odd integer, respectively), then is Schur-convex (or Schur-concave, respectively) on , and is decreasing (or increasing, respectively).
- (i)
- is Schur-geometrically convex on ;
- (ii)
- if r is even integer (or odd integer, respectively), then is Schur-geometrically convex (or Schur-geometrically concave, respectively) on .
- (i)
- is Schur-harmonically convex on ;
- (ii)
- if r is even integer (or odd integer, respectively), then is Schur-harmonically convex (or Schur-harmonically concave, respectively) on .
2. Main Results
- (i)
- is increasing on and Schur-convex on ;
- (ii)
- If , then is Schur-m-power convex on ;
- (iii)
- For , if r is even integer (or odd integer, respectively), then is Schur-m-power convex (or Schur-m-power concave, respectively) on .
- (i)
- The function is Schur-geometrically convex and Schur-harmonically convex on .
- (ii)
- If r is even integer (or odd integer, respectively), then is Schur-convex, Schur-geometric convex, and Schur-harmonic convex (or Schur-concave, Schur-geometric concave, and Schur-harmonic concave, respectively) on .
3. Some Applications
4. Conclusions
- (see Theorem 4) Let . If , then is decreasing and Schur m-power convex on .
- (see Theorem 5) Let .
- (i)
- is increasing on and Schur-convex on ;
- (ii)
- If , then is Schur-m-power convex on ;
- (iii)
- For , if r is even integer (or odd integer, respectively), then is Schur-m-power convex (or Schur-m-power concave, respectively) on .
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Shi, H.-N.; Du, W.-S. Schur-Power Convexity of a Completely Symmetric Function Dual. Symmetry 2019, 11, 897. https://doi.org/10.3390/sym11070897
Shi H-N, Du W-S. Schur-Power Convexity of a Completely Symmetric Function Dual. Symmetry. 2019; 11(7):897. https://doi.org/10.3390/sym11070897
Chicago/Turabian StyleShi, Huan-Nan, and Wei-Shih Du. 2019. "Schur-Power Convexity of a Completely Symmetric Function Dual" Symmetry 11, no. 7: 897. https://doi.org/10.3390/sym11070897
APA StyleShi, H. -N., & Du, W. -S. (2019). Schur-Power Convexity of a Completely Symmetric Function Dual. Symmetry, 11(7), 897. https://doi.org/10.3390/sym11070897