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 equation φ − 1 ( ∫ 0 1 φ ( t x + ( 1 − t ) y ) d μ ( t ) ) ⋅ ψ − 1 ( ∫ 0 1 ψ ( t x + ( 1 − t ) y ) d ν ( t ) ) = x y , 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 ] .