### Abstract

Symmetric cone (SC) monotone functions and SC-convex functions are real scalar valued functions which induce Löwner operators associated with a simple Euclidean Jordan algebra to preserve the monotone order and convex order, respectively. In this paper, for a general simple Euclidean Jordan algebra except for octonion case, we show that the SC-monotonicity (respectively, SC-convexity) of order r is implied by the matrix monotonicity (respectively, matrix convexity) of some fixed order r' (≥ r). As a consequence, we draw the conclusion that (except for octonion case) a function is SC-monotone (respectively, SC-convex) if and only if it is matrix monotone (respectively, matrix convex).

Original language | English |
---|---|

Pages (from-to) | 499-512 |

Number of pages | 14 |

Journal | Journal of Nonlinear and Convex Analysis |

Volume | 17 |

Issue number | 3 |

Publication status | Published - 2016 |

### Fingerprint

### Keywords

- Euclidean Jordan algebra
- Lowner operator
- Matrix-monotone
- SC-convex
- SC-monotone
- Symmetric cone

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology
- Control and Optimization
- Applied Mathematics

### Cite this

*Journal of Nonlinear and Convex Analysis*,

*17*(3), 499-512.

**Symmetric cone monotone functions and symmetric cone convex functions.** / Chang, Yu Lin; Chen, Jein Shan; Pan, Shaohua.

Research output: Contribution to journal › Article

*Journal of Nonlinear and Convex Analysis*, vol. 17, no. 3, pp. 499-512.

}

TY - JOUR

T1 - Symmetric cone monotone functions and symmetric cone convex functions

AU - Chang, Yu Lin

AU - Chen, Jein Shan

AU - Pan, Shaohua

PY - 2016

Y1 - 2016

N2 - Symmetric cone (SC) monotone functions and SC-convex functions are real scalar valued functions which induce Löwner operators associated with a simple Euclidean Jordan algebra to preserve the monotone order and convex order, respectively. In this paper, for a general simple Euclidean Jordan algebra except for octonion case, we show that the SC-monotonicity (respectively, SC-convexity) of order r is implied by the matrix monotonicity (respectively, matrix convexity) of some fixed order r' (≥ r). As a consequence, we draw the conclusion that (except for octonion case) a function is SC-monotone (respectively, SC-convex) if and only if it is matrix monotone (respectively, matrix convex).

AB - Symmetric cone (SC) monotone functions and SC-convex functions are real scalar valued functions which induce Löwner operators associated with a simple Euclidean Jordan algebra to preserve the monotone order and convex order, respectively. In this paper, for a general simple Euclidean Jordan algebra except for octonion case, we show that the SC-monotonicity (respectively, SC-convexity) of order r is implied by the matrix monotonicity (respectively, matrix convexity) of some fixed order r' (≥ r). As a consequence, we draw the conclusion that (except for octonion case) a function is SC-monotone (respectively, SC-convex) if and only if it is matrix monotone (respectively, matrix convex).

KW - Euclidean Jordan algebra

KW - Lowner operator

KW - Matrix-monotone

KW - SC-convex

KW - SC-monotone

KW - Symmetric cone

UR - http://www.scopus.com/inward/record.url?scp=85014028540&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014028540&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85014028540

VL - 17

SP - 499

EP - 512

JO - Journal of Nonlinear and Convex Analysis

JF - Journal of Nonlinear and Convex Analysis

SN - 1345-4773

IS - 3

ER -