An invariance of geometric mean with respect to generalized quasi-arithmetic means

In this paper, we study the invariance of the geometric mean with respect to some generalized quasi-arithmetic means, namely, we present some results concerning the functional equationφ−1(∫01φ(tx+(1−t)y)dμ(t))⋅ψ−1(∫01ψ(tx+(1−t)y)dν(t))=xy,x,y∈I, where I is an open real interval, φ,ψ:I→R are continuous strictly monotonic unknown functions and μ, ν are Borel probability measures on the interval [0,1].