Given a weighted mean \(M\) on positive reals and vector of weights \((\lambda_n)_{n=1}^\infty\). We are searching for the best constant \(C_{M,\lambda}\) which satisfies the property \[\lambda_1x_1+\lambda_2M\big((x_1,x_2),(\lambda_1,\lambda_2)\big)+\lambda_3M\big((x_1,x_2,x_3),(\lambda_1,\lambda_2,\lambda_3)\big)+\dots \le C_{M,\lambda}\big(\lambda_1x_1+\lambda_2x_2+\lambda_3x_3+\dots\big)\] for all vectors \(x=(x_1,\dots)\) of positive reals.

For a given continuous, strictly monotone function \(f\colon I \to \mathbb{R}\) (\(I\) is an subinterval of \(\mathbb{R}\)) we define the quasiarithmetic mean \(A_f\) by \[ A_f(x_1,\dots,x_n)=f^{-1}\bigg( \frac{f(x_1)+\dots+f(x_n)}{n}\bigg). \] This family, axiomatized in 1930s by Kolmogorov, Nagumo, and de Finetti plays an important role in both functional equations and economy.

A quasideviation \(E \colon I \times I \to I\) is a function which meets the properties:
(i) for all \(x,u\in I\), the equality \(\text{sign} E(x,u)=\text{sign}(x-u)\) holds;
(ii) for all \(x\in I\), the mapping \(I\ni u\mapsto E(x,u)\) is continuous;
(iii) for all \(x,y\in I\) with \(x < y\), the mapping \((x,y)\ni u\mapsto\frac{E(x,u)}{E(y,u)}\) is strictly decreasing.
For a quasideviation \(E\) on \(I\), for all \(n\in\mathbb{N}\) and \(x_1,\dots,x_n\in I\), we define a \(E\)-quasideviation mean of vector \(x\) as the element \(u\in I\) such that \[ E(x_1,u)+\dots+E(x_n,u)=0.\]