MSC 2010: 03E72, 26E50, 28E10In this paper, we prove a Stolarsky type inequality for pseudo-integrals.

We work with a fixed N-tuple of quasi-arithmetic means $M₁,...,M_N$ generated by an N-tuple of continuous monotone functions $f₁,...,f_N:I\to ℝ$ (I an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^N\ni b\mapsto (M₁\left(b\right),...,M_N\left(b\right))$ tend pointwise to a mapping having values on the diagonal of $I^N$. Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M₁,...,M_N$ taken on b. We effectively...