Quasi-Cauchy quotients and means

Let $$I\subset {\mathbb {R}}$$ I ⊂ R be an interval that is closed under addition, and $$ k\in {\mathbb {N}}$$ k ∈ N , $$k\ge 2\,$$ k ≥ 2 . For a function $$f:I\rightarrow \left( 0,\infty \right) $$ f : I → 0 , ∞ such that $$F\left( x\right) :=\frac{f\left( kx\right) }{ kf\left( x\right) }$$ F x : = f k x k f x is invertible in I, the k-variable function $$ M_{f}:I^{k}\rightarrow I,$$ M f : I k → I , $$\begin{aligned} M_{f}\left( x_{1},\ldots ,x_{k}\right) :=F^{-1}\left( \frac{f\left( x_{1}+\cdots +x_{k}\right) }{f\left( x_{1}\right) +\cdots +f\left( x_{k}\right) } \right) , \end{aligned}$$ M f x 1 , … , x k : = F - 1 f x 1 + ⋯ + x k f x 1 + ⋯ + f x k , is a premean in I, and it is referred to as a quasi Cauchy quotient of the additive type of generator f. Three classes of means of this type generated by the exponential, logarithmic, and power functions, are examined. The suitable quasi Cauchy quotients of the exponential types (for continuous additive, logarithmic, and power functions) are considered. When I is closed under multiplication, the quasi Cauchy quotient means of logarithmic and multiplicative type are studied. The equalities of premeans within each of these classes are discussed and some open problems are proposed.