# Barbour path functions and related operator means

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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 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.

Original language English 2434-2441 8 Linear Algebra and Its Applications 439 8 https://doi.org/10.1016/j.laa.2013.06.035 Published - Oct 15 2013 Yes

### Fingerprint

Operator Mean
Mathematical operators
Path
Integral Means
Order Relation
Geometric mean
Monotone Operator
Exponential functions
Fractional
Interpolate
Approximation

### 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

In: Linear Algebra and Its Applications, Vol. 439, No. 8, 15.10.2013, p. 2434-2441.

Research output: Contribution to journalArticle

title = "Barbour path functions and related operator means",
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 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.",
keywords = "Operator mean, Operator monotone function, Path",
author = "Noboru Nakamura",
year = "2013",
month = "10",
day = "15",
doi = "10.1016/j.laa.2013.06.035",
language = "English",
volume = "439",
pages = "2434--2441",
journal = "Linear Algebra and Its Applications",
issn = "0024-3795",
publisher = "Elsevier Inc.",
number = "8",

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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

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U2 - 10.1016/j.laa.2013.06.035

DO - 10.1016/j.laa.2013.06.035

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EP - 2441

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

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