Generalizations of Several Inequalities Related to Multivariate Geometric Means
Abstract
:1. Introduction
2. Definitions and Lemmas
- 1.
- If
- 2.
- For and , if
- 1.
- If on Ω implies , then we say that the function is Schur convex on Ω.
- 2.
- If is a Schur convex function on Ω, then we say that φ is Schur concave on Ω.
- 1.
- If for and , then Ω is called a geometrically-convex set.
- 2.
- If Ω is a geometrically-convex set and
- 3.
- If is a Schur geometrically-convex function on Ω, then φ is said to be a Schur geometrically-concave function on Ω.
- 1.
- If , the function is increasing on ; if , the function is decreasing on ; if , the function is increasing on and decreasing on .
- 2.
- If , the function is increasing on ; if , the function is decreasing on and increasing on .
- 3.
- For , the function is decreasing on .
- 4.
- If , the function is increasing on ; if and , the function is decreasing on .
3. Main Results
- if , then is Schur concave on ;
- if , then is Schur convex on ;
- if , then is Schur geometrically convex on ;
- if , the is Schur geometrically concave on .
- if , then is Schur geometrically convex on ;
- if , then is Schur geometrically concave on .
- if , then is Schur convex on ;
- if , then is Schur concave on ;
- if and ,
- (a)
- when , we have ;
- (b)
- when ,
- by Lagrange’s mean value theorem, we have
- from Newton’s inequality, we obtain
- 1.
- If , , and , or if , , and , or if , , and , then
- 2.
- If , , and , or if , , and , or if , , and , then
- 3.
- If , , and or if , , and , then
- 4.
- If , , and , then
- if , then is Schur concave on ;
- if , then is Schur convex on ;
- if , then is Schur concave on ;
- if , then is Schur convex on ;
- if , then is Schur geometrically convex on ;
- if , then is Schur geometrically concave on ;
- if , then is Schur geometrically convex on .
- if , then is Schur convex on ;
- if , then is Schur concave on ;
- if and ,
- (a)
- when , we have ;
- (b)
- when , using Cauchy’s mean value theorem, we have
- if , then is Schur geometrically convex on ;
- if , then is Schur geometrically concave on ;
- if , then is Schur geometrically convex on .
- 1.
- If , , and , or if , , and , or if , , and , then
- 2.
- If , , and or f , , and , then
- 3.
- If , is an even integer, and , or if , k is an even integer, , and , then
- if , then is Schur concave on ; if and n is an even (or odd, respectively) integer, then is Schur concave (or Schur convex, respectively) on ;
- if , then is Schur geometrically convex on ; if , then is Schur geometrically concave on ; if and n is an even (or odd, respectively) integer, then is Schur geometrically concave (or convex, respectively) on .
- if , then for , so the function is a Schur-concave function on ;
- if and k is an even (or odd, respectively) integer, then for . This shows from Lemma 1 that, if and n is an even (or odd, respectively) integer, the function is Schur concave (or Schur convex, respectively) on ;
- if and , from (28), it follows that
- if , the function is Schur concave on ;
- if and k is an even (or odd, respectively) integer, the function is Schur concave (or Schur convex, respectively) on ;
- if , the function is Schur concave on .
- if , then for ;
- if and k is an even (or odd, respectively) integer, then for ;
- if , then
- if , the function is Schur geometrically convex on ;
- if and k is an even (or odd, respectively) integer, the function is a Schur geometrically-concave (or convex, respectively) function on ;
- if , the function is Schur geometrically concave on .
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Xi, B.-Y.; Wu, Y.; Shi, H.-N.; Qi, F. Generalizations of Several Inequalities Related to Multivariate Geometric Means. Mathematics 2019, 7, 552. https://doi.org/10.3390/math7060552
Xi B-Y, Wu Y, Shi H-N, Qi F. Generalizations of Several Inequalities Related to Multivariate Geometric Means. Mathematics. 2019; 7(6):552. https://doi.org/10.3390/math7060552
Chicago/Turabian StyleXi, Bo-Yan, Ying Wu, Huan-Nan Shi, and Feng Qi. 2019. "Generalizations of Several Inequalities Related to Multivariate Geometric Means" Mathematics 7, no. 6: 552. https://doi.org/10.3390/math7060552
APA StyleXi, B.-Y., Wu, Y., Shi, H.-N., & Qi, F. (2019). Generalizations of Several Inequalities Related to Multivariate Geometric Means. Mathematics, 7(6), 552. https://doi.org/10.3390/math7060552