Barbour path functions and related operator means

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−xbx (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.