New characterizations of some mean-values
\textit{H. Haruki} proved in Publ. Math. 38, No. 3/4, 323-332 (1991; Zbl 0746.39004) among others that \(p^{-1} (\int_0^{2\pi} p(s)dt/(2\pi))\) (\(p\) is a real valued \(C^2\) function on the positive real and \(s=(a^2\cos^2t+ b^2\sin^2t)^{1/2})\) equals the arithmetic or geometric mean or the root-mean-square of \(a,b\) (no weights) if and only if \(p(s)= A\log s+B\), \(=As^{-2}+B\) or \(=As^2+B\), respectively. In the present paper the authors first choose \(s=a\sin^2t+b\cos^2t\) and offer that in this case the above integral mean equals the arithmetic or geometric mean or the square-mean-root or the geometric mean of the harmonic and of the geometric mean of \(a,b\) iff \(q(s)= As+B\), \(=As^{-1}+B\), \(=A\log s+B\) or \(=As^{-2}+B\), respectively. Similar results are offered for \(s=(a^{-1} \sin^2t+b^{-1}\cos^2t)^{-1}\). Reviewer's remark: In Theorem 2(iii) + should stand in place of \(-\). Also, in the introduction the authors state ``postulates'' for mean values \(M\) of two variables \(a,b\). One of the postulates requires that \(M\) is defined for all positive \(a,b\), the second that \(M\) be a symmetric function, the third that \(M(a,a)=a\). Surprisingly, it is not postulated that the value \(M(a,b)\) lie between those of \(a\) and \(b\) (strictly or otherwise).
scientific article; zbMATH DE number 930256