Means and averages of Taylor polynomials (Q1260874)

Let \(f\) have a continuous \((r+ 1)\)st derivative, nowhere 0 on \([a,b]\) and \(P_ c\) its Taylor polynomial at \(c\). Noting that \(P_ a(M)= P_ b(M)\), if \(r\) is odd, and \(2f(M)= P_ a(M)+ P_ b(M)\), if \(r\) is even, have unique solutions, the author denotes these by \(M^ r_ p\) if \(f(x)= x^ p\) (creating some confusion with his previous notation \(M^ r_ f\)). He observes that \(M^ r_{r+1}\), \(M^ r_{r/2}\), and \(M^ r_{-1}\) are the arithmetic, geometric, and harmonic means, respectively, of \(a\) and \(b\). However, an argument is offered showing that the generalization, via general probability measures, of this definition to mean values of \(n\) numbers never yields the arithmetic, geometric or harmonic means. Finally, the author makes a correction to his 1990 paper in the same Journal 149, No. 1, 220-235 (1990; Zbl 0706.26019).