A note on generalized Heronian means. (English) Zbl 1128.26302

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.


26D20 Other analytical inequalities
26E60 Means
Full Text: DOI