On the local and global comparison of generalized Bajraktarevi\'c means

Given two continuous functions $f,g:I\to\mathbb{R}$ such that $g$ is positive and $f/g$ is strictly monotone, a measurable space $(T,A)$, a measurable family of $d$-variable means $m: I^d\times T\to I$, and a probability measure $\mu$ on the measurable sets $A$, the $d$-variable mean $M_{f,g,m;\mu}:I^d\to I$ is defined by $$M_{f,g,m;\mu}(\pmb{x}) :=\left(\frac{f}{g}\right)^{-1}\left( \frac{\int_T f\big(m(x_1,\dots,x_d,t)\big) d\mu(t)} {\int_T g\big(m(x_1,\dots,x_d,t)\big) d\mu(t)}\right) \qquad(\pmb{x}=(x_1,\dots,x_d)\in I^d).$$ The aim of this paper is to study the local and global comparison problem of these means, i.e., to find conditions for the generating functions $(f,g)$ and $(h,k)$, for the families of means $m$ and $n$, and for the measures $\mu,\nu$ such that the comparison inequality $$M_{f,g,m;\mu}(\pmb{x})\leq M_{h,k,n;\nu}(\pmb{x}) \qquad(\pmb{x}\in I^d)$$ be satisfied

Production functions having the CES property

To what measure does the CES (constant elasticity of substitution) property determine production functions? We show that it is not possible to find explicitely all two variable production functions f(x; y) having the CES property. This slightly generalizes the result of R. Sato [16]. We show that if a production function is a quasi-sum then the CES property determines only the functional forms of the inner functions, the outer functions being arbitrary (satisfying some regularity properties). If in addition to CES property homogeneity (of some degree) is required then the (twovariable) production function is either CD or ACMS production function. This generalizes the result of [4] and also makes their proof more transparent (in the special case of degree 1 homogeneity)