Quasi-Cauchy quotients and means (Q2055252)

This paper contains the study of quasi-Cauchy quotients of different types. If \(I\subseteq \mathbb{R}\) is an interval (closed under addition or multiplications, respectively) and \(k\in\mathbb{N}\) with \(k\geq 2\), \(f:I\rightarrow(0,\infty)\), \(F(x)=\frac{f(T(x,\ldots,x))}{L(f(x),\ldots,f(x))}\) then \(M_f(x_1,\ldots,x_k)=F^{-1}\left(\frac{f(T(x_1,\ldots,x_k))}{L(f(x_1),\ldots,f(x_k))}\right)\) is a quasi-Cauchy quotient. If \(T\) and \(L\) are addition, then \(M_f\) is of additive type, if \(T\) and \(L\) are multiplication, \(M_f\) is of multiplicative type, if \(T\) is addition \(L\) is multiplication then \(M_f\) is of exponential type, and if \(T\) is multiplication and \(L\) is addition, then \(M_f\) is of logarithmic type. The author studies these quasi-Cauchy quotients for various type of functions, showing when they are premeans on means and poses the question when \(M_f=M_g\). For additive type as the author points out, continuous additive functions \(f\) have no meaning but the cases where \(f\) is of exponential (new class of means), logarithmic (single premean) and multiplicative type (no premeans) are studied. Similar results are obtained for the other types of quasi-Cauchy quotients.