On the intersection of two-parameter mean value families (Q2716109)

Given two sets of distinct real numbers \(r,s\) and \(p,q\), the equal weight Gini and extended, or Stolarsky, means of a positive \(n\)-tuple \(\underline a\) are, respectively, NEWLINE\[NEWLINEG_n^{r,s}( \underline a)=\left({{\sum_{i=1}^n a_i^r}\over{\sum_{i=1}^na_i^s}}\right)^{1/(r-s)} \quad \text{and} \quad E^{p,q}(\underline a)=\left({{(q-n+2)_{n-1}\sum_{i=1}^n a_i^p \big/Q'(a_i)}\over{(p-n+2)_{n-1}\sum_{i=1}^n a_i^q \big/Q'(a_i)}}\right)^{1/(p-q)},NEWLINE\]NEWLINE where \(Q(x) =\prod_{i=1}^n(x-a_i)\) and \((a)_b\) is the Pochhammer symbol \(\Gamma(a+b)/ \Gamma(a)\). The values when \(r=s\), \(p=q\), are obtained by taking limits [see \textit{C. Gini}, Metron 13, No. 2, 3-22 (1938; Zbl 0018.41404) and \textit{K. B. Stolarsky}, Math. Mag. 48, 87-92 (1975; Zbl 0302.26003)]. It has been known for some time that in the case \(n=2\) with \(s=r-1\) that these two families of means have only the arithmetic, geometric and harmonic means in common. NEWLINENEWLINENEWLINEThis very interesting paper completely solves this problem for all cases as follows; if \(n=2\) the families of means have in common only the power means; if \(n>2\) the only means in common are the arithmetic, geometric and harmonic means. In particular since all power means are particular cases of the Gini means this implies that the only power means that are extended means when \(n>2\) are the three classical means, a conjecture of \textit{E. B. Leach} and \textit{M. C. Sholander} [J. Math. Anal. Appl. 92, 207-223 (1983; Zbl 0517.26007)]. The proof depends on the properties of the two functions \(g_{r,s} (z)= G_n^{r,s}(e^z,1,\ldots, 1)^{(r-s)}\), \(f_{p,q} (z)=E_n^{p,q}(e^z,1,\ldots, 1)^{(p-q)}\), \(z\in \mathbb C\), and finally, as in the proofs of the original results by \textit{H. W. Gould} and \textit{M. E. Mays} [J. Math. Anal. Appl. 101, 611-621 (1984; Zbl 0582.41026)], on comparing the power series expansions of \( E_n^{p,q}(e^z,1,\ldots, 1)\) and \( G_n^{r,s}(e^z,1,\ldots, 1)\). The calculations in the general case are much more complicated and are done with the aid of a computer programme.