A note on generalized Heronian means (Q2745654)

The mean \(H(a,b)= {a+ \sqrt{ab}+ b\over 3}\) is called the Heronian mean and is of course just \({2\over 3}A(a,b) +{1\over 3}G(a,b) \). It is natural to consider the generalized Heronian mean \(H^{[t]}(a,b)=(1-t)A(a,b) +tG(a,b),\,0\leq t\leq 1 \), and to ask for the best possible \(p,q\) such that \(M^{[p]}(a,b)\leq H^{[t]}(a,b)\leq M^{[q]}(a,b)\), where \(M^{[p]}(a,b)\) denotes the power mean. In the classical case, \(t= 1/3\), the answer was given by the present author and Alzer. Putting \(t= w/(w+2),\,w\geq 0\), the author shows that \( p = \log 2/\log(w+2),\,q= 2/(w+2)\). In a similar manner best exponents \(s,t\) are obtained for the inequalities \(H^{[s]}(a,b)\leq M(a,b)\leq H^{[t]}(a,b)\) where \(M\) is either the logarithmic or identric mean. It would be of some interest to know the origin of the name Heronian mean.