Barbour path functions and related operator means

Noboru Nakamura

doi:10.1016/j.laa.2013.06.035

Barbour path functions and related operator means

Noboru Nakamura

Research output: Contribution to journal › Article › peer-review

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 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	https://doi.org/10.1016/j.laa.2013.06.035
Publication status	Published - Oct 15 2013

Keywords

Operator mean
Operator monotone function
Path

ASJC Scopus subject areas

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

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10.1016/j.laa.2013.06.035

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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.",

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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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Barbour path functions and related operator means

Abstract

Keywords

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