An invariance of the geometric mean with respect to Stolarsky mean-type mappings
Let \(E_{r,s} (x,y)\) be the Stolarsky mean of \(x,y> 0\). Let \(G\) be the geometric mean. The authors prove that \(G(E_{r, s}(x,y), E_{k,m} (x, y))= G(x,y)\) if and only if one of the following conditions occur: (i) \(k=m =r= s= 0\); (ii) \(k= -r\), \(m= s= 0\); (iii) \(k= m\neq 0\), \(r= s\neq 0\) and \(k=- r\); (iv) \(rk\neq 0\), \(r\neq s\), \(k\neq m\) and either \(r=-s\) and \(k= -m\), or \(k=-r\) and \(m=-s\), or \(k=-s\) and \(m=-r\). As an application, they study the functional equation \(F(x,y) =F(M(x,y), N(x,y))\), where \(M\) and \(N\) are some means, a special case of which was a problem posed by H. Haruki and Th. M. Rassias.
scientific article; zbMATH DE number 2002339
scientific article