### 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 |
---|---|

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 |

### Fingerprint

### Keywords

- Operator mean
- Operator monotone function
- Path

### ASJC Scopus subject areas

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

### Cite this

**Barbour path functions and related operator means.** / Nakamura, Noboru.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 439, no. 8, pp. 2434-2441. https://doi.org/10.1016/j.laa.2013.06.035

}

TY - JOUR

T1 - Barbour path functions and related operator means

AU - Nakamura, Noboru

PY - 2013/10/15

Y1 - 2013/10/15

N2 - 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 Gx(a,b) at x=0,12 and 1. If a=1 and b=t, then both the functions Fx(1,t) and Gx(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.

AB - 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 Gx(a,b) at x=0,12 and 1. If a=1 and b=t, then both the functions Fx(1,t) and Gx(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.

KW - Operator mean

KW - Operator monotone function

KW - Path

UR - http://www.scopus.com/inward/record.url?scp=84882876729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84882876729&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2013.06.035

DO - 10.1016/j.laa.2013.06.035

M3 - Article

AN - SCOPUS:84882876729

VL - 439

SP - 2434

EP - 2441

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 8

ER -