### 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 |
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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 Jan 1 |

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### 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.