### Abstract

We consider Barbour path function ^{Fx}(a,b)=a×bax+ba(1-x) x+ba(1-x) (0≤x≤1, a,b>0) as an approximation of the exponential function (or the geometric mean path) ^{Gx}(a,b)=a1^{-x} ^{bx} (0≤x≤1, a,b>0) by a linear fractional function, which interpolates G_{x}(a,b) at x=0,12 and 1. If a=1 and b=t, then both the functions F_{x}(1,t) and G_{x}(1,t) are operator monotone in t, parameterized with x. We also consider the order relation between the integral mean for the Barbour path function and another mean.

Original language | English |
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Pages (from-to) | 2434-2441 |

Number of pages | 8 |

Journal | Linear Algebra and Its Applications |

Volume | 439 |

Issue number | 8 |

DOIs | |

Publication status | Published - Oct 15 2013 |

Externally published | Yes |

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### Keywords

- Operator mean
- Operator monotone function
- Path

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Numerical Analysis