Some reproducing identities for families of mean values (Q1090872)

The authors determine the parameters \(t_ k\) \((k=1,2,3,4)\) for which \(M_{t_ 1}(M_{t_ 2}(x,y),M_{t_ 3}(x,y))=M_{t_ 4}(x,y)\) for all \(x>0\), \(y>0\), where either \(M_{t_ k}(x,y)=(x^{t_ k}+y^{t_ k})/(x^{t_ k-1}+y^{t_ k-1})\) \((k=1,2,3,4)\) or \(M_{t_ k}(x,y)=((x^{t_ k}+y^{t_ k})/2)^{1/t_ k}\) if \(t_ k\neq 0\) and \(=(xy)^{1/2}\) if \(t_ k=0\) \((k=1,2,3,4)\).